Computer simulations of the dynamic properties of methane in a model silica gel
Molecular dynamics (MD) simulations are reported for a Lennard-Jones fluid adsorbed into a model silica gel to study the dynamic properties of the adsorbed methane molecules. The mean-square displacement and velocity autocorrelation function of the adsorbed molecules are calculated for a set of...
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Інститут фізики конденсованих систем НАН України
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| Цитувати: | Computer simulations of the dynamic properties of methane in a model silica gel / T. Patsahan, A. Trokhymchuk, M. Holovko // Condensed Matter Physics. — 2003. — Т. 6, № 1(33). — С. 3-21. — Бібліогр.: 21 назв. — англ. |
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irk-123456789-1206912017-06-13T03:03:23Z Computer simulations of the dynamic properties of methane in a model silica gel Patsahan, T. Trokhymchuk, A. Holovko, M. Molecular dynamics (MD) simulations are reported for a Lennard-Jones fluid adsorbed into a model silica gel to study the dynamic properties of the adsorbed methane molecules. The mean-square displacement and velocity autocorrelation function of the adsorbed molecules are calculated for a set of supercritical temperatures at low (gas-like) and high (liquid-like) fluid densities and compared with the same data for a bulk fluid. The evaluated radial distribution functions illustrate the formation of a contact layer on the pore surface that is consistent with the decrease in the mobility of the adsorbed molecules in a porous environment. The calculated self-diffusion coefficient indicates a good quantitative agreement with the measured data for methane confined to the silica gel. Було здійснено комп’ютерне моделювання методом молекулярної динаміки (МД) для Ленарда-Джонсівського флюїда адсорбованого в модельному силікагелі. Обчислено функції середньо-квадратичного відхилення координат та автокореляційні функції швидкостей частинок флюїда при малих та великих густинах. Пораховані радіальні функції розподілу ілюструють явище формування контактного шару на поверхні пор, яке приводить до зменшення рухливості молекул, адсорбованих в пористому середовищі. Отримані значення для коефіцієнта самодифузії добре узгоджуються із експериментальними даними для метану, адсорбованому в силікагелі. 2003 Article Computer simulations of the dynamic properties of methane in a model silica gel / T. Patsahan, A. Trokhymchuk, M. Holovko // Condensed Matter Physics. — 2003. — Т. 6, № 1(33). — С. 3-21. — Бібліогр.: 21 назв. — англ. 1607-324X PACS: 02.70.Ns, 45.20.Dd, 51.20.+d, 61.43.Gt DOI:10.5488/CMP.6.1.3 http://dspace.nbuv.gov.ua/handle/123456789/120691 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| language |
English |
| description |
Molecular dynamics (MD) simulations are reported for a Lennard-Jones
fluid adsorbed into a model silica gel to study the dynamic properties of the
adsorbed methane molecules. The mean-square displacement and velocity
autocorrelation function of the adsorbed molecules are calculated for a
set of supercritical temperatures at low (gas-like) and high (liquid-like) fluid
densities and compared with the same data for a bulk fluid. The evaluated
radial distribution functions illustrate the formation of a contact layer on
the pore surface that is consistent with the decrease in the mobility of the
adsorbed molecules in a porous environment. The calculated self-diffusion
coefficient indicates a good quantitative agreement with the measured data
for methane confined to the silica gel. |
| format |
Article |
| author |
Patsahan, T. Trokhymchuk, A. Holovko, M. |
| spellingShingle |
Patsahan, T. Trokhymchuk, A. Holovko, M. Computer simulations of the dynamic properties of methane in a model silica gel Condensed Matter Physics |
| author_facet |
Patsahan, T. Trokhymchuk, A. Holovko, M. |
| author_sort |
Patsahan, T. |
| title |
Computer simulations of the dynamic properties of methane in a model silica gel |
| title_short |
Computer simulations of the dynamic properties of methane in a model silica gel |
| title_full |
Computer simulations of the dynamic properties of methane in a model silica gel |
| title_fullStr |
Computer simulations of the dynamic properties of methane in a model silica gel |
| title_full_unstemmed |
Computer simulations of the dynamic properties of methane in a model silica gel |
| title_sort |
computer simulations of the dynamic properties of methane in a model silica gel |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2003 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/120691 |
| citation_txt |
Computer simulations of the dynamic
properties of methane in a model silica
gel / T. Patsahan, A. Trokhymchuk, M. Holovko // Condensed Matter Physics. — 2003. — Т. 6, № 1(33). — С. 3-21. — Бібліогр.: 21 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT patsahant computersimulationsofthedynamicpropertiesofmethaneinamodelsilicagel AT trokhymchuka computersimulationsofthedynamicpropertiesofmethaneinamodelsilicagel AT holovkom computersimulationsofthedynamicpropertiesofmethaneinamodelsilicagel |
| first_indexed |
2025-07-08T18:24:52Z |
| last_indexed |
2025-07-08T18:24:52Z |
| _version_ |
1837104201937715200 |
| fulltext |
Condensed Matter Physics, 2003, Vol. 6, No. 1(33), pp. 3–21
Computer simulations of the dynamic
properties of methane in a model silica
gel
T.Patsahan∗1 , A.Trokhymchuk 1 , M.Holovko 1,2
1 Institute for Condensed Matter Physics
of the National Academy of Sciences of Ukraine,
1 Svientsitskii Str., 79011 Lviv, Ukraine
2 Ivan Franko Lviv National University,
12 Dragomanov Str., 79005 Lviv, Ukraine
Received November 20, 2002
Molecular dynamics (MD) simulations are reported for a Lennard-Jones
fluid adsorbed into a model silica gel to study the dynamic properties of the
adsorbed methane molecules. The mean-square displacement and veloc-
ity autocorrelation function of the adsorbed molecules are calculated for a
set of supercritical temperatures at low (gas-like) and high (liquid-like) fluid
densities and compared with the same data for a bulk fluid. The evaluat-
ed radial distribution functions illustrate the formation of a contact layer on
the pore surface that is consistent with the decrease in the mobility of the
adsorbed molecules in a porous environment. The calculated self-diffusion
coefficient indicates a good quantitative agreement with the measured data
for methane confined to the silica gel.
Key words: fluid, porous medium, diffusion, simulation, molecular
dynamics.
PACS: 02.70.Ns, 45.20.Dd, 51.20.+d, 61.43.Gt
1. Introduction
The knowledge of the properties of the sorbate in specific porous solids is of
great interest to both fundamental and applied science. Porous materials being ex-
tensively used as catalysts and adsorbents in the chemical industry appear to be a
powerful stimulus for statistical-mechanics studies. There are two major features of
the porous materials that affect the dynamics and other properties of the adsorbed
fluids. These features are related to the structure of the porous confinement and to
the degree of interaction of the sorbent molecules with the pore surfaces, i.e. they
∗E-mail: tarpa@icmp.lviv.ua
c© T.Patsahan, A.Trokhymchuk, M.Holovko 3
T.Patsahan, A.Trokhymchuk, M.Holovko
are related to wetting or non-wetting phenomena. The first one can be recognized
as the geometrical aspect of the effect of the porous environment on the sorbate
while the second one can be associated with the physical and/or chemical aspect of
the adsorption process. A number of theoretical studies have been accomplished for
the last two decades to model and investigate the physical adsorption into porous
media. The researchers have primarily been concerned with highly idealized single
pore spaces (slit-like, cylindrical and spherical pores).
Recently, the models for more complex porous microstructures that character-
ize silica Vycor glasses and silica gels have been developed [1,2] to study various
properties of adsorbed fluids. These models are aimed at accounting for the effect of
the structure and the topology of the pore space upon the fluid behavior in porous
materials. The model of Vycor glass proposed by Rovere et al. [1] and utilized in a
number of recent papers [3–5] is a single pore model which is quite acceptable due to
the low porosity of this kind glasses where the interaction between fluid molecules
from different pores can be neglected while the details of the pore structure can be
taken into account. In particular case, considering the hydrophilic groups placed on
the surface in porous silica-like Vycor glass, Rovere et al. [1] modelled the system
on a site-site level that includes the interactions between the atoms of the adsorbed
molecules and the pore material. Computer simulations based on such a model give
quite realistic results for the structural and dynamic properties of the fluid adsorbed
into the porous silica Vycor glass. However, in the case of silica gel we deal with a
random porous structure. If for silica gel we use a model similar to that for silica
glass [1], taking into account the randomness of pore volume distribution in the silica
gels, the number of interaction sites will increase critically leading to an incredible
increase of computing time. Therefore, one may prefer simplified models where the
fluid-surface interaction is replaced with some effective potential.
A relatively simple but still nontrivial model for simple fluids adsorbed into a
heterogeneous porous solid was formulated by Kaminsky and Monson (KM) [2].
Although KM model is simpler than the one used by Rovere et al. [1], it differs in
principle from the previously studied models [6–8] for the adsorption of hard-sphere-
like fluids into hard-sphere porous media (hard-sphere matrix). The attractive fluid-
matrix and fluid-fluid interactions are included in the KM model. Hence, the effect
of temperature on the properties of an adsorbed fluid is presented. The fluid-matrix
attraction is much stronger compared with the fluid-fluid interaction. Moreover, this
model is also characterized by quite a large asymmetry of the diameters of matrix
obstacles and of the adsorbed fluid molecules. The main attention in the studies
based on the KM model has been paid so far to the structure and thermodynamic
properties. To illustrate the utility of the KM model, the calculation of the Henry’s
law constant for methane adsorbed in a silica gel has been performed [2]. However,
little attention has been paid to the fluid dynamics in heterogeneous microstructures.
The asymmetry of energies and diameters may result in a strong accumulation of the
fluid molecules in the vicinity of matrix particles. This has already been confirmed
by Monte Carlo (MC) computer simulations [2,8]. Thus, we can expect that the
mobility of fluid molecules could also be affected.
4
Dynamic properties of methane in silica gel
The theoretical tools to investigate porous systems include the integral equation
theory approaches [9–11] and computer simulation techniques [2,6–8]. Since in the
present study our main goal is to investigate how the dynamic properties of the ad-
sorbed fluid are altered under heterogeneous porous solid confinement, we applied
molecular dynamics (MD) simulation technique. MD simulations are a versatile tool
for a detailed description of the time evolution of the simulated system, simultane-
ously giving access to the evaluation of both structural and dynamic properties. In
particular, we calculated the radial distribution functions, the running coordination
numbers of fluid molecules around the matrix particles, mean square displacements,
velocity autocorrelation functions and molecular self-diffusion in a porous medium
trying to distinguish between the geometrical and interaction contributions. The
remainder of the paper is organized as follows. The next section 2 describes the
potentials of the fluid-fluid and the fluid-matrix interactions and the method used
in the simulations. The results and the discussion are presented in section 3. Finally,
the conclusions are drawn in section 4.
2. Modelling and computer simulations
2.1. Model description
The bulk fluid is modelled by the truncated and shifted 12–6 Lennard-Jones
potential:
uff(r) =
φLJ(r) − φLJ(Rf), r < Rf
0, r > Rf
, (1)
where Rf is the cut-off distance for the fluid-fluid interaction and
φLJ(r) = 4ε
[
(
σ
r
)12
−
(
σ
r
)6
]
, (2)
with the parameters: ε/k = 148.2 K, σ = 0.3817 nm that correspond to the methane-
methane interaction. The Lennard Jones diameter σ is used as a unit of length scale
through the whole paper. The bulk fluid is characterized by the reduced number
density, ρ?
f = Nfσ
3/V , where Nf is the number of fluid molecules and V is the
volume occupied by the fluid system.
The model for a porous medium (or a porous matrix) is reduced to an equilibrium
configuration of Nm large hard spheres (obstacles) of the diameter D. The ratio of
the volume occupied by the matrix obstacles to the volume V of the porous sample
is η = (π/6)NmD3/V and defines the porosity of the matrix, 1 − η. It has already
been shown that this model is a reasonable first approximation to the structure of
silica gel [12].
The essence of the modelling of the fluid adsorbed into a porous medium is
the interaction between the fluid molecules and the matrix particles. In the present
study this is based on the model developed by Kaminsky and Monson (KM) [2] for a
methane adsorbed in a porous silica gel. According to the KM model an interaction
5
T.Patsahan, A.Trokhymchuk, M.Holovko
between the fluid molecules and the spherical matrix obstacles has the following
form:
φKM(r) =
2
3
πρsD
3εs
[
(r6 + 21/20D2r4 + 3/16D4r2 + D6/192)d12
(r2 − D2/4)9
−
d6
(r2 − D2/4)3
]
(3)
with the parameters: d = 0.33 nm, ρs = 44 nm−3, εs/k = 339 K and D = 7.055 σ. We
employed this potential function to define two matrix models useful for the purposes
of the present study. The first of them is equivalent to the KM model and we call it
an attractive matrix due to the attractive nature of the fluid-matrix interaction:
uATT
fm
(r) =
∞, r < 1
2
D
φKM(r) − φKM(Rm), 1
2
D < r < Rm
0, r > Rm
. (4)
Additionally, we introduce the repulsive matrix which is characterized by the repul-
sive potential of interaction between fluid molecules and matrix particles:
uREP
fm
(r) =
∞, r < 1
2
D
φKM(r) − φKM(Rm), 1
2
D < r < R0
0, r > R0
. (5)
This last potential represents the repulsive part of the KM model interaction where
the parameter R0 = 4.1526 σ was chosen in the way to ensure that the difference
φKM(r = R0) − φKM(Rm) is equal to zero.
The long-range attractive potential functions for the fluid-fluid and the fluid-
matrix interactions, (1) and (4), both have been spherically truncated with the
cut-off radii, Rf and Rm, respectively. To eliminate the impulsive contribution to
the force from the discontinuity of the potential, both truncated potentials have
been shifted by the magnitude of the interaction energy at the cut-off distances. In
the present study we used Rf = 2.56 σ for the fluid-fluid interaction and Rm = 12 σ
for the fluid-matrix interaction.
2.2. Molecular dynamics simulations
The molecular dynamics (MD) simulations of the fluid adsorbed into a porous
medium were performed through the adoption of commonly used velocity Verlet
algorithm [13]. The modifications we introduced into this algorithm were caused
by the presence in the simulation cell of the two quite distinct components, i.e.
mobile fluid molecules and large static matrix particles. Thereby, we should solve
the equations of motion exclusively for the fluid molecules, similarly to the case of
the bulk fluid, but with the obstacles taken into account. For such a complex system,
care has to be taken with respect to the total momentum which could not be equal
6
Dynamic properties of methane in silica gel
to zero. To ensure the zero of the momentum in our algorithm, we monitored the
total momentum of the system as a function of time and have observed only the
negligible fluctuations around the zero value which did not effect the generated
data. In general, the properties of the fluid in a porous environment are determined
by the interaction within the fluid molecules, and by the interaction of the fluid
molecules with the static obstacles. However, the role of a fluid-matrix interaction
is frequently simplified and reduced to the effects such as geometrical obstructions,
hydrodynamic drag, etc. To shed more light on the role of a fluid-matrix interaction
and reveal the impact of geometrical and temperature effects on the behavior of a
matrix fluid, four different systems (from A through D) have been considered and
studied. The system A corresponds to the bulk fluid, i.e., Nf fluid molecules occupy
a volume V . Each of the systems B and C is composed of the same (like system A)
number of fluid molecules, Nf . However, these molecules are infused into the two
topologically identical porous media: one medium has an attractive fluid-matrix
interaction (system B) while the second one has a repulsive fluid-matrix interaction
(system C). Finally, the system D again is a bulk fluid (similar to the system A), but
with an effective fluid density, i.e., the same Nf fluid molecules are placed into an
effective volume, V ′ = V (1 − η), which corresponds to the free volume in a porous
medium in the systems B and C. The system D should be considered in order to
show an effect of the presence of the matrix obstacles when the data for a matrix
fluid (systems B and C) are compared against the bulk fluid (system A).
To choose the temperature region, we exploit the fact that in practice the adsorp-
tion into silica gels is usually observed at supercritical temperatures with respect
to the bulk fluid. To ensure the single phase conditions, the reduced temperature,
T ? = kT/ε, has been varied within the range from T ? = 1.2 to T ? = 2.0 , which is
above the bulk gas/liquid critical temperature ( T ?
c ∼ 1.1 ) for the considered fluid
model [15]. Hence, the temperature range T ? = 1.2 − 2.0 will be supercritical for
the fluid adsorbed into a porous medium as well [16]. During the simulation runs,
the average values of the desired temperatures were maintained using the velocity
rescaling procedure according to the weak-coupling scheme due to Berendsen [14]
with a coupling time constant τ ∗ ∼ 0.1 ps while the simulation time step ∆t, was set
to 2 ps. The production runs were started after the equilibration period of 440 ps.
All the matrix simulations (systems B and C) reported in the present study
have been carried out for a fixed number of matrix particles, Nm = 32, which
have been set randomly in the cubic box with the basic size L = V 1/3. This size
is determined by the porosity of the matrix, 1 − η , which was fixed at the value
corresponding to the KM model where η = 0.386. The number of fluid molecules, Nf ,
has been chosen to simulate the low (typical gas-like) and the high (typical liquid-
like) fluid densities and was fixed at Nf = 585 (in some runs 456) and Nf = 5385
molecules, respectively. For each set of the fixed parameters of the adsorbed fluid,
i.e. temperature and density, we carried out up to five different configurations of the
matrix particles. The data that are reported have been obtained by the averaging
over these configurations.
7
T.Patsahan, A.Trokhymchuk, M.Holovko
3. Results and discussion
3.1. Radial distribution functions
Figure 1. Attractive fluid-matrix inter-
action. Fluid-fluid and fluid-matrix ra-
dial distribution functions, gff(r) and
gfm(r), respectively, for a low fluid densi-
ty at the temperatures T ∗ = 2.0 (a) and
1.2 (b). Solid lines correspond to the da-
ta obtained from MD simulations of this
work while circles are GCMC simulation
data by Vega et al [8].
Although our main goal is the dy-
namic properties, we begin with the
discussion of the fluid-fluid and the
fluid-matrix radial distribution func-
tions (RDFs), gff(r) and gfm(r), respec-
tively. There were a few reasons that
we persecuted by calculating and ana-
lyzing the RDFs. One of them was to
verify the modified MD algorithm em-
ployed in the present study through the
comparison of the obtained MD data
against those computed by Vega et al.
[8] from grand canonical Monte Carlo
(GCMC) simulations. The second rea-
son was to get conclusions about the ef-
fect of the porous environment on the
fluid-fluid RDF through the comparison
with the bulk data at the same temper-
ature and density conditions. Further,
we were curious about the local order-
ing of fluid molecules near the pore sur-
faces which could be revealed from the
analysis of the fluid-matrix RDF. Final-
ly, we intended to use the information on
the structure ordering of the molecules
of the adsorbed fluid while interpreting
their dynamic properties.
The main results for the RDFs are
summarized in figures 1–3. The simu-
lation data are presented for the two
reduced temperatures, T ∗ = 1.2 and
2.0, which correspond to bottom and
top boundaries of the temperature range
where the dynamic properties have been
evaluated. Additionally, these simulation data show how the temperature affects the
structural properties of the low density (gas-like) and the high density (liquid-like)
fluids when they are adsorbed into a porous medium. First of all, for both densities
(figures 1 and 2) we found an excellent agreement between MD data and those gen-
erated from GCMC by Vega et al. [8]. Small discrepancies should be attributed to
the details of the truncation of potential functions. Analyzing the fluid-matrix RDFs
for an attractive matrix, we observe the layered structure of the fluid molecules with
8
Dynamic properties of methane in silica gel
Figure 2. The same as in figure 1 but for
a high fluid density.
a well-defined contact layer around the
matrix particles at both densities of
the adsorbed fluid. When the density
is fixed and the temperature is falling,
the peaks of fluid-matrix RDFs increase
as expected. However, at a fixed tem-
perature, the same peaks decrease with
the increase of the density of the matrix
fluid, i.e. going from gas-like to liquid-
like adsorbate. Opposite trends occur in
the case of a repulsive matrix (figure 3)
where the contact value of the fluid-
matrix RDFs increases with the increase
of the density, as expected. No layering
of fluid molecules is observed for a gas-
like adsorbate in the case of a repulsive
matrix. To illustrate the formation of a
contact layer, in figure 4 we present the
snapshot of the MD configuration of the
fluid molecules around the matrix parti-
cles for the case of an attractive matrix.
Using the fluid-matrix RDFs one can
estimate the average numbers of the flu-
id molecules in a contact layer. To do
this, we calculated the running coordi-
nation number,
n(r) = 4πρ?
f
∫ r
0
gfm(x)x2dx , (6)
for the fluid molecules adsorbed on the matrix particle. We can define the first co-
ordination number, n1, as the value of n(r) at the distance r that corresponds to
the position of the first minima of gfm(r). Corresponding data are collected in table 1.
Table 1. The number of fluid molecules in the contact layer around the matrix
particle
attractive matrix repulsive matrix
T ? ρ?
f
rmin n1 rmin n1
1.2 0.0384 4.95 34 5.0 13
0.3534 4.7 141 4.8 115
2.0 0.0299 5.15 23 – –
0.3534 4.75 139 – –
9
T.Patsahan, A.Trokhymchuk, M.Holovko
Figure 3. Comparison of the attractive
and the repulsive fluid-matrix interac-
tions. Fluid-fluid and fluid-matrix radial
distribution functions, gff(r) and gfm(r),
at temperature T ∗ = 1.2 for a low flu-
id density (a) and high fluid density (b).
Data are obtained from MD simulations
of this work. Solid lines correspond to
the data for an attractive fluid-matrix
interaction (KM model), circles – for a
repulsive fluid-matrix interaction.
One can see that the number of flu-
id molecules surrounding the matrix
particles increases with the increase of
the fluid density. The first coordination
number has a very small dependence on
the temperature for a high fluid density.
3.2. Mean square displacement and
velocity autocorrelation function
The main subjects of our interest
from the point of view of dynamic prop-
erties are two dynamic functions. The
first is the time evolution of the mean
square displacement (MSD), as defined
by
〈r2(t)〉 =
1
Nf
〈
∑
i
|ri(t) − ri(0)|2
〉
,
(7)
where the ri(t) are the space coordi-
nates of the center of mass of the fluid
molecules at time t. The second dynam-
ic function which we evaluated during
simulation runs was the normalized ve-
locity autocorrelation function (VACF),
ϕ(t) =
∑
i 〈vi(t)vi(0)〉
∑
i 〈vi(0)vi(0)〉
, (8)
with vi(t) being the individual veloc-
ities of the fluid molecules at time t.
The average 〈. . .〉 is along the trajec-
tory of each fluid molecule within the
whole ensemble and over the time. The
trajectory information can be employed
to compute the time dependent diffusiv-
ity, D(t). This can be evaluated either
using the Green-Kubo expression [17],
D(t) =
1
3
∫ t
0
ϕ(t′)dt′ , (9)
or the Einstein relation [17],
D(t) =
1
6
∂〈r2(t)〉
∂t
. (10)
10
Dynamic properties of methane in silica gel
Figure 4. The snapshot of MD configuration of the fluid molecules around the
matrix particles at a high fluid density, ρ?
f
= 0.3534 and at temperature, T ∗ = 1.2.
Only fluid molecules belonging to the contact layers around the matrix particles
are shown.
In general, we can summarize that the total length of the simulated trajecto-
ry was different for gas-like and for liquid-like fluids as well as it was different for
bulk and for matrix fluids. Longer trajectories were needed in order to achieve a
good statistical description of the MSD at a low density. At the fixed fluid densi-
ty, final trajectories which correspond to the macroscopic self-diffusion coefficient
D = D(t → ∞) are shorter when the fluid is adsorbed in a porous medium. Rela-
tively shorter trajectories (comparatively to MSD) are needed for recording VACF
to ensure their decay to the final value of zero (within the statistical accuracy of
the simulation data). Generally speaking, the random structure of a porous medium
may lead to the anisotropy of the diffusion process. However, we were interested in
the average features of the molecule dynamics only.
The set of MSDs as a function of the observation time for the low and the high
fluid densities are shown in figures 5a and 5b, respectively. Each figure contains
two groups of curves which correspond to the bulk fluid (dashed lines) and to the
matrix fluid (solid lines) under the same temperature conditions. As it is already
known from the bulk studies, the molecules have a much bigger displacement for
the low fluid density than for the high fluid density. The MSD is also bigger at a
higher temperature. The presence of the matrix particles leads to the slowing down
of the diffusion motion: fluid molecules diffuse at considerably shorter distances in
the porous medium than in the bulk conditions. This is observed for both densities
and for all temperatures studied.
11
T.Patsahan, A.Trokhymchuk, M.Holovko
Figure 5. MD simulation data for the mean square displacement. (a): Comparison
of the bulk and matrix fluids at different temperatures, T ∗ = 1.2; 1.4; 1.6; 1.8; 2.0
(from the bottom to the top) and a for low fluid density; (b): The same as in
(a) but for a high fluid density; (c): Comparison of the bulk fluid, free volume
fluid and matrix fluid with attractive and repulsive fluid-matrix interactions at
temperature T ∗ = 1.2 for a low fluid density; (d): The same as in (c) but for a
high fluid density.
12
Dynamic properties of methane in silica gel
The time dependence of the MSD is different at different periods of time. Three
different time regimes have been distinguished by analyzing the movement of the
tracer from the time dependence of the correlation function of the scattered-light
intensity [18,19]: (i) the diffusion process is normal at short times, (ii) becomes
anomalous at moderate times and (iii) returns to normal for large delay times.
Normal diffusion assumes here that time dependence of the fluid molecule dynamics
is subject to the law of deterministic motion (Newtonian dynamics) on the initial
ballistic phase and satisfies the Fick’s second law and isotropy (Euclidean dynamics)
on a long time scale. The analysis of the simulation data on this subject is presented
in figures 5c and 5d for the low and the high fluid densities, respectively. In order
to extract the information from the simulation data, the log-log plots of MSD are
used.
Two linear regions (at the initial and final stage) can be easily recognized for
each set of the data presented on the plots in figures 5c and 5d. In general, every
linear region on logarithmic scale indicates a distinct diffusive regime, described by
a power law behavior D(t) ∼ tα where the power α can be determined from the
slope of the straight line fitting the data. Below the time, t? = t(ε/mσ2)1/2 ∼ 1,
the straight line has the slope α = 2, corresponding to Newtonian dynamics, i.e.
〈r2(t)〉 = v2(0)t2. Well above t? ∼ 1, the straight line has the slope α = 1 corre-
sponding to Einsteinian regime, i.e. 〈r2(t)〉 ∼ Dt. In the case of a bulk fluid, these
two straight lines intersect, defining the characteristic distance λ and time τ that
separate the two diffusive regimes. Between these two regimes, the displacement pro-
file is not linear, representing the transition from deterministic (quasi-free) molecule
motion to the Einsteinian diffusion (molecule motion is affected by the collisions
with the other molecules). Comparing the MSD curves for the bulk fluid with the
MSD curves for the gas-like matrix fluid (figure 5c) we see that, indeed, as it was
found experimentally in the studies of probe diffusion through polyacrylamide gels
[18], the shape of the MSD profile has an additional linear region in between two
normal diffusion regimes. The slope of this linear region for a low density adsorbate
is about α = 3/2 indicating a perturbation of the molecule trajectory due to the col-
lision with the obstacles. This behavior only quantitatively depends on the nature of
the surface of the matrix particles (attractive or repulsive fluid-matrix interaction).
For an attractive matrix, the transition time from ballistic behavior to anomalous
diffusion is shorter as well as the crossover from anomalous diffusion to the normal
diffusion over long distances is more delayed. After entering the anomalous diffusion
regime, MSD is permanently larger for repulsive obstacles.
Anomalous diffusion is strongly affected by the density of the adsorbed fluid
(figure 5d). In particular, the linear region at moderate times practically does not
exist for the liquid-like adsorbate, i.e. the molecules are found to follow the normal
(Brownian) diffusion directly after the initial ballistic phase. As in the case of a
low fluid density, time dependence of the MSD is the same for both attractive and
repulsive matrix particles. The explanation to this behavior is that at a high density
of the adsorbed fluid, the surface coverage of matrix particles is very high for both
(repulsive and attractive) matrices (see the first coordination numbers in table 1) and
13
T.Patsahan, A.Trokhymchuk, M.Holovko
Figure 6. MD simulation data for normalized velocity autocorrelation function.
(a): Comparison between the matrix fluids at low and high fluid densities and at
different temperatures, T ∗ = 1.2; 1.4; 1.6; 1.8; 2.0 (from the bottom to the top);
(b): Comparison of the bulk fluid, free volume fluid and matrix fluid with attrac-
tive and repulsive fluid-matrix interactions at temperature T ∗ = 1.2 for a low
fluid density; (c): The same as in (b) but for a high fluid density; (d): The same
as in (b) but in log scale; (e): The same as in (c) but in log scale.
14
Dynamic properties of methane in silica gel
the tracer cannot “see” the naked surface of the matrix obstacles but can “see” only
the fluid molecules from the adsorbed (or contact) layer. Due to this, the diffusion
process in the porous medium at a high density of the adsorbed fluid is qualitatively
similar to the diffusion of the bulk (free volume or effective) fluid at a higher density.
A noticeable difference in the dynamic properties of gas-like and liquid-like ma-
trix fluids is also evidenced by the data simulated for the VACFs which are shown in
figure 6. For each fluid density the set of five curves are displayed (see figure 6a) to
illustrate the temperature effect. The both groups of VACFs relax following the qual-
itatively different patterns, which means that the motions of individual molecules are
essentially different. It is notable that ϕ(t) relaxes much faster in the case of a high
density adsorbate and assumes the negative values at t? ∼ 1 . The slow relaxation
at a low fluid density is superimposed by a weak non-periodic oscillating pattern.
The normalized VACFs, evaluated for bulk, for free volume, and for both (attrac-
tive and repulsive) matrix fluids are shown in figures 6b–6e for both the gas-like
and the liquid-like fluids. One can see, that ϕ(t) of the matrix fluids (both repulsive
and attractive) decays faster than that in the bulk fluid, regardless of time. The
deviation is observed starting from the times t? ∼ 0.2 and appears to be significant
at the times greater than the time t?
con
roughly required for the molecules to diffuse
into contact with the matrix particles. In contrast to the bulk fluid, the well defined
minima in ϕ(t) profile for the case of a matrix fluid are seen at short times.
3.3. Macroscopic self-diffusion coefficient
Both expressions, equations (9) and (10), can be used to predict the macroscopic
self-diffusion coefficient, D = D(t → ∞) . However, in both cases the functions ϕ(t)
and 〈r2(t)〉 have to be known at the large times compared to the characteristic time
of correlations between the tagged molecule and its immediate neighbors. At these
times, which determine the hydrodynamic region, we can expect that 〈r2〉 behaves
like a linear function of t . Hence, D can be calculated as the limiting slope of the
MSD profile. In such a context an evaluation of the self-diffusion coefficient, D ,
through the VACF is more rigorous. This is illustrated in figure 7 where the MD
simulation data are displayed for the time-dependent self-diffusion, D(t) , evaluated
from both the VACFs and the MSDs and for all four systems discussed in the
present study. One can see that for the bulk as well as for the matrix fluids both
roots lead to consistent results (within each simulated system) at large times. In
the bulk case, both procedures give consistent results with a similar uncertainty
which is of the order of 5% [20]. The same seems to be valid for the matrix fluid:
the final values of self-diffusion as t → ∞ are equal being calculated from MSD
or VACF. However, these final values are different for each simulated system; the
difference between attractive and repulsive matrix vanishes for a high fluid density.
An interesting observation that follows from the data shown in figure 7 concerns the
maxima for self-diffusion versus time in the case of both matrix fluids that is clearly
seen when the log scale is used (figures 7c and 7d). This is an important qualitative
difference between the bulk and the matrix fluids and can serve as another evidence
of the possible anomalies in the diffusion processes in a porous environment.
15
T.Patsahan, A.Trokhymchuk, M.Holovko
Figure 7. Time-dependent self-diffusion at temperature T ∗ = 1.2. Solid lines cor-
respond to the results obtained from the MSD data while dashed lines correspond
to the results evaluated from the VACF data. (a): Comparison of the bulk flu-
id, free volume fluid and matrix fluids with attractive and repulsive fluid-matrix
interactions for low fluid density; (b): The same as in (a) but for a high fluid
density; (c): The same as in (a) but in log scale; (d): The same as in (b) but in
log scale.
16
Dynamic properties of methane in silica gel
Figure 8. Relative self-diffusion co-
efficient of the attractive matrix
[Dmatr/Dbulk] and the effective (free
volume) [Deff/Dbulk] fluids versus tem-
perature T ∗, for the low and the high
fluid densities. The meaning of solid and
dashed lines is the same as in figure 7.
Figure 9. Logarithmic self-diffusion
coefficient of the attractive matrix
fluid versus 1/T ∗ for the low and the
high fluid densities. The meaning of
solid and dashed lines is the same as
in figure 7.
The simulation data for the macroscopic self-diffusion coefficient, D , are sum-
marized in figures 8–10. Two sets of data for each temperature and density case
correspond to the self-diffusion coefficient obtained from the MSDs (solid lines) and
from the VACFs (dashed lines). We see, that in all cases the agreement between both
procedures is quite reasonable. The ratios of the diffusion coefficients of the attrac-
tive matrix fluid (system B) and the bulk fluid with an effective density (system D)
to the corresponding quantity for the bulk conditions (system A), Dmatr/Dbulk and
Deff/Dbulk, respectively, clearly show that Dmatr changes relatively to both the bulk
fluid and the bulk fluid with an effective density (figure 8). This means, that the
diffusion processes in a porous environment are significantly affected not only by the
decrease of the volume but also by the presence of the matrix obstacles; diffusion is
considerably slower in this particular case of a porous medium. This effect is more
pronounced for the gas-like adsorbate where Dmatr is one order of magnitude lower
than for the free volume case, Deff , while for a high density of the adsorbed fluid one
can see a half of the order of magnitude decrease. Examining these data, we see that
most of the decrease in Dmatr is caused by the presence of a porous confinement.
The substitution of the matrix fluid by the bulk fluid with the effective density that,
indeed, is associated with the decrease of diffusion, does not correspond and is not
equivalent to the effect of a porous confinement. Not only the magnitudes of the
diffusion coefficients Deff and Dmatr are different (Dmatr is always lower than Deff
at the same fluid density and temperature), but the dependence on temperature
and density in both cases is qualitatively different: ratio Dmatr/Dbulk increases with
17
T.Patsahan, A.Trokhymchuk, M.Holovko
Figure 10. Self-diffusion coefficient of the adsorbed methane molecules in the
silica gel versus temperature T in real units for gas-like and liquid-like fluid den-
sities. Square symbols with a solid line correspond to the results obtained from
the MSD data, while circles with a dashed line correspond to the results evaluat-
ed from the VACF data. The filled symbols correspond to the experimental data
for methane and ethane gases in silica gel.
the rise of temperature while Deff/Dbulk decreases; ratio Dmatr/Dbulk for a low fluid
density is smaller than for a higher fluid density while Deff/Dbulk has an opposite
trend.
The effect of temperature on the diffusivity can be seen in figure 9. This plot
shows that the diffusion of gas-like and liquid-like adsorbates in a porous medium
is an activated process, i.e. it follows the Arrhenius equation,
D(T ) = D0 exp(−Ea/RT ) , (11)
where D0 is called the preexponential factor, Ea is the activation energy, R is the
gas constant, and T is the temperature. The estimate of an activation energy is
3.53 kJ/mol for low a density of fluid molecules and 1.80 kJ/mol for the fluid
molecules at a liquid-like fluid density.
The absolute values of the prediction for the self-diffusion coefficient for the
methane adsorbed in silica gel are shown in figure 10. From this plot one can also
see the dependence of the self-diffusion coefficient Dmatr on the temperature: with
the rise of the temperature, the diffusion increases and is accelerated at a low fluid
density relative to high fluid density. Our estimates, Dmatr = 0.875 · 10−7 m2/s, for
the gas-like fluid density at T = 296 K are in a reasonable agreement with the
experimental value 0.952 · 10−7 m2/s for the pore diffusivity of the methane gas in
silica gel at T = 310 K [21].
18
Dynamic properties of methane in silica gel
4. Conclusions
Summarizing, the MD simulations are applied while investigating the dynamic
properties of simple fluids adsorbed in a porous medium. The study is based on the
model for the adsorption of methane in silica gel developed by Kaminsky and Monson
[2]. MD simulations that we have performed allow us to examine the dependence of
the dynamic properties of the adsorbed fluid on: (i) the presence of static obstacles
that form a random porous structure; (ii) specific (attractive or repulsive) interaction
between the fluid molecules and the matrix particles; (iii) the density of the adsorbed
fluid; (iv) the temperature conditions in the system.
To analyze the dynamic properties, both the velocity autocorrelation functions
and the mean square displacements have been evaluated. Comparing the MSD curves
for the bulk fluid with the MSD curves for the matrix fluid we found that the
shape of the MSD profile has an additional linear region in between two normal
diffusion regimes. The slope of this linear region for a low density adsorbate is about
α = 3/2 indicating a perturbation of the molecule trajectory due to collision with
the obstacles. This behavior qualitatively agrees with experimental findings in the
studies of probe diffusion through the polyacrylamide gels [18]. Such an anomaly in
the MSD versus time is not practically seen for a high density of the matrix fluid
since the surface coverage of the matrix particles by the fluid molecules is very high
and the tracer cannot “see” the naked surface of the matrix obstacles but it can
“see” only the fluid molecules from the adsorbed layer. Due to this, the diffusion
process in a porous medium at a high density of the adsorbed fluid is qualitatively
similar to the diffusion of the bulk fluid at a higher density.
Both the velocity autocorrelation functions and the mean square displacements
have been used to predict the macroscopic self-diffusion coefficient, D . It was noticed
that the self-diffusion coefficient is more convenient to be obtained from the velocity
autocorrelation function. The presence of the matrix particles affect the mobility of
fluid molecules. At a low density of the matrix fluid, D strongly depends on the
temperature. The effect of temperature decreases when the density of the adsorbed
fluid increases. Moreover, we have shown that the diffusion of both gas-like and
liquid-like adsorbates in a porous medium is an activated process, i.e. it follows the
Arrhenius equation. The self-diffusion coefficient is smaller for a matrix fluid than
for a bulk fluid. It is shown, that D decreases not only due to the decrease of the
space available (an excluded volume effect) for adsorption but also due to the specific
geometry of the confinement. Comparison of the calculated self-diffusion coefficient
with the measured data for the methane confined to the silica gel indicates a good
quantitative agreement.
References
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20
Dynamic properties of methane in silica gel
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