ARTICLE IN PRESS
Physica A ] (]]]]) ]]]–]]]
www.elsevier.com/locate/physa
Thermal diffusion in micropores by molecular
dynamics computer simulations
Guillaume Galliéroa,b,, Jean Colombanic, Philippe A. Boppd,
Bernard Duguayd, Jean-Paul Caltagironeb, Franc- ois Montele
a
LFC (UMR-5150), Université de Pau et des Pays de l’Adour, Centre Universitaire de recherche scientifique,
BP 1155, F-64013 Pau Cedex, France
b
Laboratoire TREFLE (UMR-8508), site ENSCPB, Université Bordeaux I, 16 Avenue Pey-Berland,
F-33607 Pessac Cedex, France
c
LPMCN (UMR-5586), Université Claude Bernard Lyon I, 6 rue Ampère,
F-69622 Villeurbanne Cedex, France
d
LPCM (UMR-5803), Université Bordeaux I, 351, Cours de la Libération, F-33405 Talence Cedex, France
e
TOTAL, CSTJF, Avenue Laribbau, F-64018 Pau, France
Abstract
This work focuses on the identification of the main microscopic processes that influence
thermal diffusion (the Soret effect) in a fluid mixture confined in an uncorrugated slit pore. To
achieve this purpose, a boundary driven nonequilibrium molecular dynamics scheme is applied
on binary mixtures of super-critical Lennard–Jones (LJ) spheres representing methane and ndecane. Following previous work, we perform a systematic study of the influence of the
parameters used to describe a model slit pore on an effective thermal diffusion factor. Among
these parameters are: The nature of the reflection of the diffusing particles on the walls
(specular or diffusive), the pore width with respect to the particle size and the fluid-wall
potential strength. Simulations were run both on equimolar and non-equimolar mixtures.
The results indicate that thermal diffusion is effectively lowered only for strong fluid–wall
interactions. It is shown that the general trends, which are different under sub- and
Corresponding author. LFC (UMR-5150), Université de Pau et des Pays de l’Adour, Centre
Universitaire de recherche scientifique, BP 1155, F-64013 Pau Cedex, France. Tel.: +33 5 59 40 76 46; fax:
+33 5 59 40 76 95.
E-mail address: [email protected] (G. Galliéro).
0378-4371/$ - see front matter r 2005 Elsevier B.V. All rights reserved.
doi:10.1016/j.physa.2005.06.001
ARTICLE IN PRESS
2
G. Galliéro et al. / Physica A ] (]]]]) ]]]–]]]
super-critical conditions, can be explained by a careful analysis of the relative sorption
energies of the two compounds.
r 2005 Elsevier B.V. All rights reserved.
Keywords: Thermal diffusion; Molecular dynamics; Slit pore; Lennard–Jones; Adsorption
1. Introduction
The evaluation of the initial state of a petroleum reservoir needs an accurate
description of the oil field behavior. However, the existing models sometimes fail to
describe all encountered situations [1]. One aspect of this limitation comes from the
omission of the thermal diffusion process [2]. This cross process, also called Soret
effect in condensed phases, characterizes the coupling between mass flux and
temperature gradient [3]. It is characterized by the Soret coefficient S T ¼ DT =D
(where DT is the thermal diffusion coefficient and D Fick’s diffusion coefficient) or the
thermal diffusion ratio aT ¼ TST (T being the average temperature). It is now
established that this effect, although of second order, modifies the arrangement of the
fluid components inside petroleum reservoirs [4]. Unfortunately, the physics of this
phenomenon is still not well understood, partly because only a few sets of reliable
experimental data are available in the literature [5]. Moreover, even less is known
about the influence of a porous medium on this cross effect. This is, however, a crucial
point for the description of oil reservoirs. For instance, from a phenomenological
point of view, the interdiffusion coefficient D of a binary mixture is replaced in Fick’s
law by an effective coefficient D when diffusion proceeds through a porous medium
[6,7]. The modification of D by the porosity is called tortuosity t ¼ ðD=D Þ1=2 and is of
the order of t1:35 in a typical oil field environment. Therefore, an improvement in
the basic understanding of this phenomenon would be more than welcome before it is
introduced into in the existing numerical models of reservoir fluid distribution.
Nevertheless, during the last decade, real progresses have been achieved [8]. New
experimental techniques [5,9], as well as numerical ones [10–13], seem to be able to
provide now reliable values of Soret coefficients in bulk fluids. On the other hand,
these methods and models are often limited to a particular kind of mixture and/or to
a particular thermodynamic state.
Among the possible approaches to estimate thermal diffusion, we have chosen the
molecular dynamics (MD) numerical simulation technique. This tool has been
shown to yield correct estimations of the thermal diffusion in various mixtures
[14–17]. It provides, furthermore, a detailed insight into the molecular mechanisms
underlying this process [15–18]. Moreover, and this is what has motivated this work,
it is possible to evaluate the influence of a simple porous structure on the thermal
diffusion process [19–22].
We have chosen a slit pore geometry to model the porous medium because of its
geometrical simplicity; this structure reduces the problem to a bi-dimensional one,
viz. the direction of the temperature gradient, i.e., parallel to the pore walls and the
one perpendicular to it. In this geometry, equilibrium self-diffusion has been shown
ARTICLE IN PRESS
G. Galliéro et al. / Physica A ] (]]]]) ]]]–]]]
3
to be modified by the confinement. The magnitude of this modification depends on at
least three features: the slit pore width [23], the fluid–wall interaction and the
corrugation of the walls (via the ‘‘slipping length’’ [24] at the interface for the last
two characteristics).
The transposition of equilibrium thermodynamic properties in a porous structure
to a non-equilibrium system (i.e., here subject to a thermal gradient) is not at all
obvious. For example, the macroscopic temperature and concentration gradients
imposed on the non-equilibrium fluid may very well be modified by the presence of a
solid matrix. Three main effects due to the presence of a slit pore can thus be
envisaged to disturb the thermodiffusion:
as in the equilibrium self-diffusion case, the geometry of the pore should influence
the fluid dynamics (via the pore width and/or the corrugation of the walls);
the fluid–wall interaction may alter the particle distribution in the pore because of
selective adsorption of molecules at the walls;
the thermal conductivity of the solid could modify the thermal field in the liquid.
To shed some light on these points we have performed numerical molecular
simulations, using a non-equilibrium molecular dynamics boundary driven scheme,
having in mind three objectives:
(1) From a methodological point of view, we wanted to test the validity of a
simplified model of the solid (‘‘integrated fluid–wall potential’’ [25]) with two
extreme types of particle reflection, diffusive and specular, on the pore walls. This
requires comparisons with simulations using a complete atomistic representation
of the walls.
(2) From an industrial point of view, we have chosen our molecular parameters in
order to model methane and n-decane in super-critical conditions for the liquid,
and aluminosilicate for the wall. These systems correspond to typical
components encountered in petroleum reservoirs.
(3) From a more fundamental point of view, we have focused on the Soret
steady state (characterized by aT ) where the Soret and Fick fluxes exactly
compensate. So we have left aside various effects like transient flows, relative
change of DT and D; . . .. By changing the pore width, the surface attraction
and the mole fractions of the components of the mixture, we hope to have
mimicked the major factors that might affect thermal diffusion in such
microporous structures.
2. Theory and models
A comprehensive description of the phenomenology of thermal diffusion in bulk
fluids is provided by linear irreversible thermodynamics [3], but the correct definition
of thermal diffusion in porous media is not a trivial task. This complexity comes
from the fact that when a thermal gradient is applied to a fluid confined in a porous
ARTICLE IN PRESS
G. Galliéro et al. / Physica A ] (]]]]) ]]]–]]]
4
structure, cross effects, apart from thermal diffusion, may appear: thermocapillary
convection due to interfacial tension gradients [19,21], barodiffusion due to the
pressure drop induced by the thermal gradient [26], etc. If we focus on the purely
diffusive contributions according to the general formulation of transport in porous
media [27], the molar flux of one species of a non-reacting mixture under thermal
gradient can be written in terms of five contributions: the Knudsen flow, the diffusive
transport, the viscous flow, the surface diffusion and the thermal diffusion, some of
these contributions being small in closed system. Following previous work [20], we
have considered that it is possible to gather the Knudsen, the diffusive and the
surface contributions into an overall diffusion flow characterized by an effective
diffusive coefficient De , using a definition for the thermal diffusion factor, aT ,
formally similar to the one used in bulk fluids:
aT ¼ T
rx DT
¼
,
xð1 xÞ rT
De
(1)
where T is the average temperature and x the mole fraction of the component
involved. This effective thermal diffusion factor will be denominated in the following
simply ‘‘thermal diffusion factor’’.
Throughout this work, a positive thermal diffusion factor, aT , means that methane
migrates predominantly towards the hot areas. A sketch of the simulation box is
given in Fig. 1.
Simulations are performed in mixtures composed of model-methane (CH4 )
and n-decane (C10 H22 ), described by simple Lennard–Jones (LJ) 12–6 potentials.
This united atom modeling neglects the chain structure of the decane molecule
in order to reduce computer time. Otherwise, it would not be possible to
do the extensive simulation runs needed to study the variations in composition,
pore width, etc.
Fig. 1. Scheme and orientation of the slit pore used in the simulations.
ARTICLE IN PRESS
G. Galliéro et al. / Physica A ] (]]]]) ]]]–]]]
We write the LJ potential between particles i and j as
sij 12 sij 6
U ij ¼ 4eij
,
r
r
5
(2)
where eij is the potential strength (well depth), sij the sum of the radii of the particles
i and j and r the interparticle distance. For pure compound i ¼ j. For mixtures it is
possible to define sij and eij from the sii and the eii with the set of combination rules
defined below.
In order to be consistent with previous work, we have used the potential
parameters given by Simon et al. [28], which provide a correct estimation of the
pressure–temperature phase diagram of the mixture. The cross molecular parameters
between fluid particles of different kinds have been obtained from the classical
Lorentz–Berthelot combination rules. The cut-off radius [11] in the simulations has
been taken equal to 2:5 sij .
Concerning the fluid–wall interactions, an integrated 10–4–3 Steele [25] potential
has been chosen:
" !#
4
4
10
s
s
s
2
sf
sf
sf
U slit ðyÞ ¼ 2prs esf s2sf D
,
(3)
5 y
y
3Dð0; 61D þ yÞ3
where rs is the particle density in the solid, ssf and esf are the cross molecular
parameters characterizing the interactions between fluid and hypothetical wall
particles, D is the distance between the layers of the solid and y is the distance
separating the fluid particle from the wall.
The molecular parameters of the solid have been selected to mimic the solid used
by Colombani et al. [20] as closely as possible (face-centered-cubic
aluminosilicate):
( e ¼ 1:91 kJ mol1 , D ¼ 2:245 A
( and r ¼ 0:028 A
( 3 . The Lorentz–Berthess ¼ 3 A,
s
s
lot mixing rules have again been used to calculate the parameters ssf and esf of the
fluid–solid interactions. Note that such an integrated potential, which treats the
porous medium as unstructured, affects only the components of the particles
velocities perpendicularly to the walls.
This description of the porous structure suffers from two principal weaknesses: It
does not take into account the thermal transfer inside the solid and between the solid
and the liquid. Furthermore, the corrugation inherent to the molecular nature of the
surface is omitted. Therefore, the sorption process is not well described. To partially
alleviate these deficiencies, we have introduced two different kinds of ‘‘particle
reflection’’ at the solid walls. The first one, ‘‘specular reflection’’, corresponds to
perfect slip conditions. It is the straightforward application of the Steele potential.
The second one, diffusive reflection, corresponds to a wall which adsorbs particles
and ejects them in random directions with a longitudinal velocity extracted from a
Maxwell–Boltzmann distribution at the local temperature of the wall (and a
conserved transverse velocity) [29].
It has been stated numerically that due to the large thermal conductivities of
aluminosilicates, the temperature inside the solid is nearly homogeneous along the
slit [20]. The local temperature given by the walls to the fluid has thus been fixed to
ARTICLE IN PRESS
6
G. Galliéro et al. / Physica A ] (]]]]) ]]]–]]]
the average one of the whole box. Our method is similar to the one described by
Cracknell et al. [30]. This method has the advantage to be more realistic and the
drawback to eventually create a slight transverse thermal gradient near the ends of
the box (see below).
To estimate the thermal diffusion using relation (1), we have adopted a boundary
driven non-equilibrium molecular dynamics approach. The fundamental mechanism
underlying this method is an exchange of kinetic energy between the particles located
in boundary layers. A bi-periodical thermal gradient along the x-direction can thus
be generated while keeping the periodic boundary conditions in all three directions
[11]. For this purpose, the simulation box has been divided into 32 slabs stacked in
x-direction. The fluid is heated (kinetic energy added) close to x ¼ 0 and x ¼ L
(where L is the length of the simulation box) and cooled (kinetic energy removed) at
x ¼ L=2. The reported quantities are a mean of those computed on the two
symmetric half-boxes along x; thus, only one half of the simulation box will be
shown in the figures. The slabs in which the energy redistribution mechanism is
active are discarded for the determination of the thermal diffusion factors.
This non-equilibrium algorithm provides the thermal diffusion factors directly,
i.e., without any further assumption or intervention needed in the fluid, contrary to
equilibrium or synthetic methods [14]. Moreover, this experiment-like method allows
physical processes induced by a thermal heterogeneity (mass gradient distortion,
eventual thermocapillary convectiony) to be present.
Among the existing algorithms, the heat exchange (HEX) algorithm has been
selected to generate the temperature gradient [11]. Furthermore, since we use an
integrated potential to treat the fluid–wall interactions, some mass flow due to
interfacial tension gradients close to the wall can build up. It can be seen as a
thermocapillary convection due to a microscopic Marangoni effect [19,21].
Computing locally the mass fluxes, we have verified that the imposed heat fluxes
do not yield any noticeable thermal creep.
In order to follow the law of corresponding states when changing the mole
fractions, we use the van der Waals one fluid approximation [31] to work with
reduced thermodynamics variables. Such approximation seems reasonable for the
mixtures studied [17]. The chosen reduced temperature and density, T ¼ 2:273 and
r ¼ 0:4227 (mean fluid values in the whole fluid), correspond to super-critical
conditions and have been extensively studied in a previous work [17]. For an
equimolar methane/n-decane mixture these conditions correspond to T ¼ 802:8 K
and r ¼ 0:3374 g cm3 .
One difficulty when comparing results from bulk and confined systems is the
correct definition of the thermodynamic state. Apart from the fact that in micropores
there is a known shift of the critical point [32], the problem is to estimate correctly
the volume accessible to the fluid inside the pore [19]. This is due to the repulsive part
of the fluid–wall potential: no fluid is present in the immediate vicinity of the walls.
Therefore, throughout this work, the fluid volume used to determine the density is
taken somewhat arbitrarily to be the entire volume, as determined by the centers of
the virtual wall particles, minus a volume along the surfaces with a depth equal to the
sij parameter between methane and the solid surface.
ARTICLE IN PRESS
G. Galliéro et al. / Physica A ] (]]]]) ]]]–]]]
7
To ensure sufficient statistical averaging, simulations are carried out with 1500
particles of fluid, during at least 107 time steps of 5 fs each, i.e., a total duration of
50 ns. A reduced heat flux [33] (noted J U ) of 0.085 has been applied except when
stated, which corresponds to a thermal gradient close to 20 K nm1 . The length of
the box is typically of the order of 5 nm, but change with the pore width.
The numerical parameters (cut-off radius, time step, size of the system, heat flux
imposed) are chosen to obtain the best possible compromise between simulation
times and reliability of the computed values [33].
The sampling of the various quantities has been performed on a 40 40 mesh in
the yz-plane. To analyze the repartition of the fluid particles inside the simulation
box, we have defined a reduced number density N ir of compound i per mesh unit as
N ir ¼
1600
N iDyDz ,
Ni
(4)
where N iDyDz is the average number of particles of species i per mesh unit and N i is
the number of particles i in the entire simulation box. Thus N ir is equal to one
everywhere in a homogeneous fluid.
3. Results
3.1. Preliminary simulations
First, the thermal diffusion factor has been computed for an equimolar mixture in
the super-critical thermodynamic conditions given above, without porous medium:
aT ¼ 1:40 0:06. This value will serve as a reference value to quantify the influence
of the porosity. For this particular thermodynamic state, the thermal diffusion has
been shown to be mainly caused by the modification of the n-decane local density or,
in other words, by the preferential migration of the n-decane [22,33].
We have then determined the liquid density of this mixture at the stationary state
in a 5 nm specular slit pore and, as expected, there is a strong non-homogeneity of
particle distribution perpendicular to the walls, as shown in Fig. 2.
This layering of the fluid in the vicinity of the walls is a consequence of the
repulsive part of the wall potential, which confines the liquid and induces molecular
packing [34]. The attractive part of this potential gives rise to adsorption of
molecules at the walls. This ‘‘physical bond’’ reduces the exchange probability of
molecules between the first layer and the remaining fluid. Note also that due to
thermal expansion, the density is higher in the cold region than in the hot one.
For the same system, we have then looked at the temperature distribution in
the confined fluid for specular and diffusive reflections for J U ¼ 0:085. Fig. 3 shows
that in the stationary state the thermal gradients are well established and uniform. In
the diffusive case, the thermostating influence of the walls, due to the choice of their
‘‘temperatures’’, lowers the gradient and induces a slight transverse gradient in the
y-direction.
ARTICLE IN PRESS
8
G. Galliéro et al. / Physica A ] (]]]]) ]]]–]]]
Fig. 2. Local density in a supercritical equimolar methane/n-decane mixture in a specular 5 nm wide slit
pore at the stationary state during a non-equilibrium simulation. (—) corresponds to a hot layer, (––) to a
layer at the average temperature of the system and ( ) to a cold layer.
Fig. 3. Stationary state n-decane mole fraction (up triangles, left scale) and temperature (down triangles,
right scale) in a 5 nm slit pore with diffusive (filled symbols) and specular (open symbols) boundary
conditions at the walls.
In spite of the inhomogeneity of the fluid particles distribution perpendicularly to
the walls, the progression of the mole fractions along the thermal gradient is linear,
as shown in Fig. 3. Thus, we are able to determine the thermal diffusion factor from
the so-determined temperature and mole fractions gradients.
We now concentrate on the specular walls case. When looking closer at the
diffusing molecules, some features specific to confined fluids can be distinguished.
Fig. 4 shows that the n-decane particles are much more adsorbed to the walls than
methane (i.e., the peaks close to the walls have a larger area) and that these peaks are
farther from the walls. The first feature is a consequence of the deeper potential well
for the n-decane–walls interaction and the second one of the larger radius of this
ARTICLE IN PRESS
G. Galliéro et al. / Physica A ] (]]]]) ]]]–]]]
9
Fig. 4. Reduced number density, N ir , Eq. (4), in an equimolar supercritical mixture of methane and ndecane in a 5 nm wide specular slit pore: cold and hot layers.
molecule in the united atom description. The figure indicates that, except in the first
adsorption layers, there is a slight migration of the two species toward the cold part
of the simulation box: density decreases in the hot part due to thermal expansion,
and vice- versa in the cold part.
Fig. 4 clearly indicates that, like in bulk fluid, n-decane is the component which
shows the largest reduced number density discrepancy between the hot and the cold
layers and so generates the mole fraction gradient seen in Fig. 3. This behavior is
typical for the thermodynamic state investigated (super-critical state). In normal
liquid phases (sub-critical states), the sign of thermal diffusion remains the same, but
the mole fraction gradient is induced mainly by the methane particles [20,33].
Fig. 4 shows that the thermal diffusion process takes place mostly in the central
part of the pore and much less in the adsorbed layers where the densities in the hot
and cold areas are comparable. In some sense, these particles are ‘‘frozen’’ due to
adsorption to the walls.
In addition, in order to assess the independence of the thermal diffusion factor
measured to the temperature gradient (in the linear regime), we have evaluated the
influence of the reduced heat flux imposed on aT for the same mixture in a 5 nm
specular slit pore.
Table 1 indicates that for this system the influence of the imposed heat flux on the
thermal diffusion factor remains limited. Furthermore, no general trend on the
behavior of aT with J u is noticeable, which means that, in this case, possible
disturbances, such as thermal creep, are negligible [21].
3.2. Validation of the model
As seen before, the structureless model walls that we have chosen cannot describe
all the physical processes that might occur in a microporous structure. Nevertheless,
ARTICLE IN PRESS
G. Galliéro et al. / Physica A ] (]]]]) ]]]–]]]
10
this model shows one advantage: a reduced computing time compared to an
atomistic description of the solid (which is highly favorable for these time-consuming
non-equilibrium simulations).
However, it must be checked that the physics underlying of the Soret effect is not
lost in the simplifying assumptions. Hence we have simulated with our integrated
potential and specular walls the system studied by Colombani et al. [20] with an
explicit all-atom model (i.e., all degrees of freedom of the atoms in the solid are
modeled): a methane-n-decane equimolar undercritical mixture at T ¼ 350 K and
r ¼ 0:6 gcm3 , confined in an aluminosilicate slit pore.
According to Fig. 5, both approaches show the same trends, despite slightly lower
values of aT for the integrated model than for the atomistic one. These differences
can be explained in terms of slight differences in the definition of the pore width and
finite size effects of the simulated systems. In both cases we find a weak dependency
of the thermal diffusion factor on the width of the pore for the narrowest pores
studied, whatever the wall model. For larger widths, the values of aT are close to the
one in bulk liquid. Thus, in these thermodynamic conditions, for this geometry and
this mixture, the porous medium affects the thermal diffusion process only weakly.
Table 1
Influence of the reduced heat flux on the thermal diffusion factor in an equimolar methane/n-decane
mixture confined in a 5 nm specular slit pore
J U
0.014
0.043
0.057
0.085
0.113
0.17
aT
0:8 0:13
0:99 0:12
0:97 0:1
0:88 0:1
0:9 0:08
0:92 0:08
Fig. 5. Thermal diffusion factor of an equimolar sub-critical methane/n-decane mixture in the bulk (—)
and confined in atomistic (m) and integrated specular (,) slit pores of various pore widths.
ARTICLE IN PRESS
G. Galliéro et al. / Physica A ] (]]]]) ]]]–]]]
11
Thus, contrary to mass diffusion [30,34], the integrated porous approach seems to
be a good candidate to test the influence of simple microporous structures on
thermal diffusion in a first approximation. This might be explained by the fact that
the corrugation and the thermal conductivity of the solid, not modeled by this
approach, have similar effects on the two components. Therefore, their influence on
the concentration gradients should be rather weak, especially for the shallow
potential wells (compared to kT) used here.
3.3. Equimolar mixture in a slit pore
The model being that far validated, we turn back to the confined mixture defined
in the beginning section (equimolar methane/n-decane mixture in the super-critical
regime, with T ¼ 2:273 and r ¼ 0:4227) since its thermodynamic conditions are
closer to the ones found in a petroleum field. On this mixture a systematic study of
the influence on aT of an integrated slit pore (pore width, attractiveness of the walls,
nature of the reflections) on the thermal diffusion factor has been performed.
To quantify the strength of the fluid–wall interactions we have defined a reduced
potential U slit ,
U slit ¼
U slit
,
U slitsilicate
(5)
where the reference value, U slitsilicate , has been taken equal to the one defined to mimic
the aluminosilicate solid given in [20].
We have carried out simulations, using specular slit pore, in which we have
evaluated aT for U slit ranging from 0.25 to 4 and pore widths from 3 to 15 nm, see
Fig. 6. First, we note that the thermodiffusive behavior is markedly different in subcritical (Fig. 5) and super-critical (Fig. 6) conditions with much larger variations of
aT with the pore size in the super-critical state. Secondly, Fig. 6 clearly indicates that
in all super-critical cases aT in the confined fluid is lower than in the bulk fluid.
Furthermore, the decrease of aT is enhanced by a reduction in pore width and by an
increase of U slit . Both effects have a similar influence since increasing U slit increases
the range of the walls’ influence on the fluid and so decreases the unperturbed
fluid area.
One notes that for relatively weak fluid–wall interactions (U slit p0:5), which
correspond to a fluid–wall potential depth (for n-decane) smaller than 2 kT, the
values of aT are nearly independent of the pore size and close to the one in bulk fluid.
This trend demonstrates that in this case the purely geometric confinement does not
influence the magnitude of the thermal diffusion.
To provide insight into the influence of the nature of the fluid particle reflection on
the walls, a similar set of simulations has been carried out, but using ‘‘diffusive’’
walls instead of ‘‘specular’’ ones.
The results shown in Fig. 7 indicate that the general behavior of aT is similar
to the one obtained for specular walls. However, the overall reduction of aT is less
pronounced than in specular micropores.
ARTICLE IN PRESS
G. Galliéro et al. / Physica A ] (]]]]) ]]]–]]]
12
3.4. Non-equimolar mixtures in a specular slit pore
To evaluate the influence of selective adsorption, we have then performed
simulations on non-equimolar methane/n-decane mixtures for the same super-critical
reduced thermodynamic conditions. These simulations have been performed using
specular walls.
Fig. 8 shows that, contrary to what occurs in bulk fluids [17], aT in a specular slit
pore does not increase monotonically with the mole fraction of the light component.
Fig. 6. Thermal diffusion factor of an equimolar supercritical mixture confined in a slit pore with specular
boundary conditions at the walls for various pore widths and fluid–wall potential strengths, U slit .
Bulk fluid
1.5
Thermal d iffusion factor
10 nm width
1
5 nm width
0.5
3 nm width
0
-0.5
-1
0.5
1
1.5
2
Reduced potential strength
Fig. 7. Thermal diffusion factor in an equimolar supercritical methane/n-decane mixture confined in slit
pores of various widths and with different fluid–wall interactions, U slit . Filled symbols: diffusive reflection
at the walls; open symbols: specular reflection, same results as given in Fig. 6.
ARTICLE IN PRESS
G. Galliéro et al. / Physica A ] (]]]]) ]]]–]]]
13
The deviations between bulk fluid and slit pore thermal diffusion increases with
methane mole fraction; they are amplified by a reduction of the pore size, as shown
in Fig. 8. This can even lead for the narrowest pores to a relatively constant value for
aT in the entire range of mole fractions.
Finally, we have performed simulations in a 5 nm specular slit pore for various
fluid–wall interactions, U slit , ranging from 0.25 to 2. Fig. 9 shows how the magnitude
of the thermal diffusion depends on the mole fraction and that this dependence is
strongly linked to the affinity of the fluid with the porous structure. Note that in all
cases, the aT values are smaller than in bulk fluid. Note also that, like in equimolar
mixtures, when U slit p0:5, (i.e., in weakly attractive pores), the aT values are close to
the ones obtained in bulk, whichever the mole fraction.
These results confirm what has been found previously: the greater the solid–fluid
potential, the more aT decreases. Hence the difference in aT between bulk and
confined fluid increases with increasing methane mole fraction.
4. Discussion
All numerical results are in agreement with the picture that the prevailing influence
on the magnitude (even on the sign) of the Soret effect in a slit pore (specular or
diffusive) is the strength of the adsorption of the mostly migrating species, i.e., here
of methane in sub-critical and n-decane in super-critical conditions. The adsorbed
molecules, mainly the n-decane for both thermodynamics conditions, behave like
quasi-frozen particles. Their mobility is small and they contribute only weakly to the
Soret concentration gradient.
Decreasing pore widths and increasing fluid–walls potential depths will increase
the fraction of adsorbed n-decane (the migrating species in super-critical conditions,
Fig. 8. Thermal diffusion factor in non-equimolar supercritical methane/n-decane mixtures in bulk fluid
and for two specular slit pore widths.
ARTICLE IN PRESS
14
G. Galliéro et al. / Physica A ] (]]]]) ]]]–]]]
see Figs. 4 and 10), and consequently the thermal diffusion factor decreases, as
shown in Figs. 6–9.
Thin pores and large U slit values can lead to a nearly complete adsorption of ndecane, but not of methane, see Fig. 10. Therefore, only methane molecules migrate
toward the cold part of the box and the sign of aT can be inverted. On the contrary,
when the potential minimum of U slit for n-decane is smaller than about 2 kT, which
corresponds to weakly adsorbing walls, the thermal diffusion factor is almost
independent of the pore width, see Fig. 6.
Fig. 9. Thermal diffusion factor in nonequimolar supercritical methane/n-decane mixtures confined in a
5 nm specular slit pore for various fluid-solid interaction strengths.
Fig. 10. Reduced number density, N ir , Eq. (4), in an equimolar supercritical methane/n-decane mixture
confined in a 3 nm wide diffusive slit pore, left figure, and in a 3 nm wide specular slit pore, right figure. For
both pores U slit ¼ 1. Same legend as in Fig. 4.
ARTICLE IN PRESS
G. Galliéro et al. / Physica A ] (]]]]) ]]]–]]]
15
With diffusive walls, as shown in Fig. 11, the migration of n-decane under thermal
diffusion is less hindered than with specular walls, even in the first adsorbed layer.
Therefore, the decrease of aT is weaker in diffusive slit pores than in specular ones.
In fact, by ‘‘ejecting’’ particles back toward the bulk after a contact these walls
artificially reduce the ‘‘freezing’’ effect. Hence, diffusive walls affect the magnitude of
the relative separation less than specular ones.
In super-critical non-equimolar mixtures, the n-decane mole fraction decrease
leads to a progressive vanishing of ‘‘free’’ n-decane molecules: the growing majority
of n-decane molecules are frozen. So the difference in aT , compared to bulk fluid,
increases with the mole fraction of methane, see Figs. 8 and 9. This effect, following
the trends seen in equimolar mixtures, is emphasized for thin pores (Fig. 8) and
strong fluid–wall interactions (Fig. 9) due to the increasing amount of n-decane
adsorbed. Again, if U slit for n-decane is smaller than about 2 kT, the thermal
diffusion factor is only weakly affected, whatever the mole fraction and the pore
width, see Fig. 9.
In the sub-critical states, methane is the species that migrates most and thus
creates the mole fraction gradient [22,33]. As methane is far less adsorbed than ndecane, decreasing the pore width has a weaker effect on the thermal diffusion
factor, as seen in Fig. 5.
Elsewhere [33], we noted that, to a large extent, using simple LJ spheres is known
to provide reasonable estimations for the diffusive behavior in mixtures of simple
linear alkane in the bulk. In confined systems, however, orientation effects, e.g. in the
adsorbed layers near the wall, could become important especially for narrow pores.
A more sophisticated molecular description than simple spheres is required to take
into account these possible effects [35,36].
5. Summary, conclusion and outlook
We have studied some effects on thermal diffusion due to the confinement of
simple sub- and super-critical mixtures of methane and n-decane, modeled as LJ
spheres, in a slit pore with structureless walls. Comparisons in a sub-critical state
indicate that this simple pore model seems to be able to provide relevant results for
thermal diffusion, comparable to those using an atomistic description of the pore
structure.
We have carried out a systematic analysis of the influence of the pore walls nature
(specular or diffusive reflection), of the pore width and of the fluid–wall potential
strength on equimolar and non-equimolar mixtures.
In the super-critical state, we found a general decrease of the thermal diffusion
factor in the pores, compared to the one in bulk fluid. A noticeable decrease,
however, occurs only when the maximum potential depth of fluid–wall interactions
for the n-decane is larger than about 2 kT. These effects due to the pore are enhanced
for specular reflections at the walls, compared to what occurs for diffusive ones. In
the sub-critical state, the effect of slit pore width is shown to be weak.
ARTICLE IN PRESS
G. Galliéro et al. / Physica A ] (]]]]) ]]]–]]]
16
We show that the estimation of the relative strengths of the adsorption to
the walls, combined with the knowledge of the sensitivity of the species to the
temperature gradient, allows a qualitative estimation of the modification of the
thermal diffusion factor with the slit pore characteristics. Therefore, we would
assume that in real micropores, contrary to mass diffusion, effective thermal
diffusion is essentially guided by the ratio of the specific surface over the fluid volume
and the relative affinity of each species to the solid surface. Then, knowing the
species which migrates most (the one which is the most sensitive to the temperature
gradient) and thus essentially generates the mole fraction gradient (which may be, as
shown, different for different thermodynamic states), should give a clue to predict
the general trends how the thermal diffusion factor will vary with the nature of the
porous medium.
Acknowledgements
We wish to thank Total for a grant for one of us (G.G). We thank the Centre
Informatique National de l’Enseignement Supérieur in Montpellier (France) and
the SIMOA in Bordeaux, which provided a large part of the computer time for
this study.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
B. Faissat, F. Montel, Entropie 107 (1996) 198/199.
P. Costeseque, P. Jamet, Entropie 217 (1999) 51.
S.R. de Groot, P. Mazur, Nonequilibrium Thermodynamics, Dover, New York, 1984.
M.R. Riley, A. Firoozabadi, AIChE 452 (1998) 44.
J.K. Platten, M.M. Bou-Ali, P. Costesèque, J.F. Dutrieux, G. Wittko, W. Köhler, C. Leppla,
S. Wiegand, Philos. Mag. 83 (2003) 1965.
D. Bejan, D.A. Nield, Convection in Porous Media, Springer, Amsterdam, 1998.
A.A. Shapiro, E.H. Stenby, Trans. Porous Media 41 (2000) 305.
S. Wiegand, J. Phys.: Condens. Matter 16 (2004) R357.
W. Köhler, J. Chem. Phys. 98 (1993) 660.
K. Firoozabadi, K. Ghorayeb, K. Shukla, AIChE 46 (2000) 892.
B. Hafskjold, Lect. Notes Phys. 584 (2002) 3.
L.J.T.M. Kempers, J. Chem. Phys. 115 (2001) 6330.
J. Luettmer-Strathmann, J. Chem. Phys. 119 (2003) 2892.
A. Perronace, C. Leppla, F. Leroy, B. Rousseau, S. Wiegand, J. Chem. Phys. 116 (2002) 3718.
P. Bordat, D. Reith, F. Müller-Plathe, J. Chem. Phys. 115 (2001) 8978.
B. Rousseau, C. Nieto-Draghi, J. Bonet Avalos, Europhys. Lett. 67 (2004) 976.
G. Galliero, Fluid Phase Equilibr. 224 (2004) 13.
C. Debuschewitz, W. Köhler, Phys. Rev. Lett. 87 (2001) 055901.
I. Wold, B. Hafskjold, Int. J. Thermophys. 20 (1999) 847.
J. Colombani, G. Galliero, B. Duguay, J.-P. Caltagirone, F. Montel, P.A. Bopp, Phys. Chem. Chem.
Phys. 4 (2002) 313.
G. Galliero, J. Colombani, B. Duguay, J.-P. Caltagirone, F. Montel, Entropie 239/240 (2002) 98.
J. Colombani, G. Galliero, B. Duguay, J.-P. Caltagirone, F. Montel, P.A. Bopp, Philos. Mag. 83
(2003) 2087.
ARTICLE IN PRESS
G. Galliéro et al. / Physica A ] (]]]]) ]]]–]]]
[23]
[24]
[25]
[26]
[27]
[28]
[29]
[30]
[31]
[32]
[33]
[34]
[35]
[36]
17
L. Bocquet, J.-L. Barrat, Europhys. Lett. 31 (1995) 455.
Y. Almeras, J.-L. Barrat, L. Bocquet, J. Phys. IV (Paris) 10 (2000) 27.
W.A. Steele, The Interaction of Gases with Surfaces, Pergammon Press, Oxford, 1974.
D. Nicholson, J. Membr. Sci. 129 (1997) 209.
E.A. Mason, A.P. Malinauskas, Gas Transport in Porous Media: The Dusty Gas Model, Elsevier,
Amsterdam, 1983.
J.-M. Simon, B. Rousseau, D.K. Dysthe, B. Hafskjold, Entropie 217 (1999) 29.
M. Cieplak, J. Koplik, J.R. Banavar, Physica A 274 (1999) 281.
R.F. Cracknell, D. Nicholson, K.E. Gubbins, J. Chem. Soc. Faraday Trans. 91 (1995) 1377.
J.P. Boon, S. Yip, Molecular Hydrodynamics, Dover, New York, 1991.
J. Zarragoicoechea, V.A. Kuz, Fluid Phase Equilibr. 220 (2004) 7.
G. Galliero, Ph.D. Thesis, Université Bordeaux I, 2003.
M. Schoen, Lect. Notes Phys. (1993) m17.
F. Zhang, J. Chem. Phys. 111 (1999) 9082.
J.-C. Wang, K. Fichthorn, Colloids Surfaces A 206 (2002) 267.