Molecular Dynamics Simulation of Flow in Pores
Jan Blomer
National Aerospace Laboratory, Chofu, Tokyo 182-8522, Japan
Abstract. The gaseous flow in nano-scale pores is of wide interest for many today's industrial applications,
e.g. in microelectronics, nano-mechanical devices (Knudsen compressor) and reaction and adsorption at
porous surfaces. This can be seen from a variety of papers of recent RGB Symposia. Furthermore it is
possible to separate gases by porous membranes. Although the fundamental problem of all these applications
is same, namely the important role of the gas-surface interaction in such small structures, we will primarily
concentrate on the separation of different gas species by porous membranes. These membranes are typically
very robust (temperature, chemical resistance) because they are made from ceramics which offers new
application fields. Porous flow can roughly be divided in several flow regimes by the Knudsen number:
From viscous flow to Knudsen diffusion to surface diffusion and up to capillary condensation. A Molecular
Dynamics (MD) model for the gas as well as the surface is formulated to investigate the interaction of gas
atoms or molecules with internal degrees of freedom and the pore. The MD method seems to be well suited
to study these phenomena because it can deal with the high density and the many-body-interactions, which
occur during the multilayer adsorption and condensation at the surface, although it is clear that it is limited
to a small physical space because of its high computational consumption.
I
INTRODUCTION
Gas separation by porous membranes is known since long times (Graham 1829; Knudsen 1911), although
nowerdays almost only dense membranes, which are usually made of organic materials, are used for membrane
based separation tasks because of their better selectivity. Next to the resistance, selectivity is one of the key
properties of a membrane and it is defined for a binary mixture as
2/1^2
«12 =
2/2^1
m
,
(1)
while xi is the upstream and yi the downstream molfraction and xi + x% = 1 and yi + y^ = 1. Usually cei2
is based on the better permeating component, i.e. it is larger then unity. Nevertheless porous membranes are
still interesting because they can be made from ceramics and glasses, which offers high resistance against heat
and agressive chemical environments, while organic materials are rather limited (max 250 °C). So they can be
integrated directly into chemical reactors to continously separate the products to bypass the limitations of the
thermodynamical equilibrium. The flow in pores is governed by different mechanisms. These different regimes
can be distinguished by the Knudsen number
Kn=
X
\=
l
(2)
defined as the ratio of the mean free path A of a hardsphere molecule and the pore diameter D. With increasing
Knudsen number the regimes are as follows: In large pores laminar flow occurs, which must be avoided because
no separation takes place. In smaller pores Knudsen diffusion is the main transportation mechanism. Here the
gas flows of the different components is independent from each other, because the number of gas-gas collisions
is small compared to gas-surface collisions. When the up- and downstream pressure are same, the flux [7] fii is
&kBT
CP585, Rarefied Gas Dynamics: 22nd International Symposium, edited by T. J. Bartel and M. A. Gallis
© 2001 American Institute of Physics 0-7354-0025-3/01/$18.00
531
Periodic boundary
Source
Sink
FIGURE 1. Sketch of simulation area
Usually the length of the pore is much larger than the radius, so that Eq. 3 simplifies to
Hi = -
3kBT L
(4)
The highest selectivity is reached when the downstream pressure is negligible, this leads to an ideal separation
factor ai2,id, which only depends on the mass of the components
Ml
M2'
(5)
If the interaction between the gas molecules and the pore walls is strong enough, the gas is adsorbed at the
surface wall and transported by surface diffusion. If one component is condensable, capillary condensation
might occur, i.e. a liquid drop of this component blocks the pore for the other component. This gives very
high selectivity. If the length of the pore is negligible, countereffusion can be seen, which leads to Graham's
law
HI
M2
MI
(6)
when the up- and downstream pressure is same (isobaric counterdiffusion). If the pore diameter is reduced to
the size of the permeating gas components, molecular sieving might occur, i.e. the species are separated just
by their size. All these mechanisms can occur at the same time and hinder or enhance each other. While in
this simulation only one pore of a certain diameter is used in a real membrane the poresize varies about an
average; the standard deviation of this distribution should be narrow for best separation results.
II
MOLECULAR DYNAMICS MODEL
The molecular dynamics (MD) model consists mainly of two regions: The gas-phase, which is further divided
in an upstream and downstream part, and the bulk material of the membrane (Fig. 1). The length-scale in
these two domains is usually very different: In the lattice it is determined by the binding length of the potential
and in the gas phase by the mean free path A (see Eq. 2). So usually the bulk material is much more dense
then the gas, i.e. much more atoms have to be simulated for the surface than for the gas and an effective model
for the surface has to be chosen. In this work the surface is simulated by an heat bath method based on the
Langevin differential equation as described earlier [2].
532
ENERGY
0
-2
-4
-6
-8
-10
-12
FIGURE 2.Binding
Bindingenergy
energybetween
between bath
bath and
and lattice
lattice for
for3D
3Dpart
partofofsimulation
simulationarea
area
FIGURE 2.
The pore itself is modeled by a three dimensional lattice, either by Lennard-Jones-interactions between
The
pore itself
by a three
either bylattice
Lennard-Jones-interactions
between
all atoms
withinisa modeled
cutoff distance
(2.5 A,dimensional
platinum) orlattice,
by a harmonic
with nearest and next nearest
all atoms
within
a
cuto
distance
(2.5
A
,
platinum)
or
by
a
harmonic
lattice
with
nearest
and
next
nearest
neighbour interactions (graphite). The formentioned model omits some binding energy for each lattice atom,
neighbour
interactions
model
omits some
eachbetween
lattice the
atom,
i.e. energy
caused by (graphite).
interactions The
with formentioned
the atoms outside
the cut-off
radiusbinding
and the energy
binding for
energy
i.e. atoms
energyand
caused
byatoms
interactions
with
theoutside
atomsthe
outside
the cut-o
radiusenergies
and theare
binding
energyonce
between
lattice
wich are
even
simulated
area. These
calculated
before the
atoms
latticeand
atoms
even
outside
thebysimulated
energiesparticles,
are calculated
onceform
before
the and
simulation
are wich
given are
to the
lattice
atoms
interactionarea.
with These
the heat-bath
to virtually
the ansimulation
and areThis
givenprocedure
to the lattice
by interaction
with
particles,
to virtually
form
infinite lattice.
resultsatoms
in a weak
coupling to
thethe
heatheat-bath
bath at the
inner pore
walls and
increasing
influence
outer regions.
energy for
poreheat
frombath
platinum
in Fig.
2. and
an innite
lattice.
Thisto the
procedure
resultsThe
in acoupling
weak coupling
toathe
at theis shown
inner pore
walls
increasing
inuence
to the outer regions.
couplingregions
energyonly
for aa pore
from platinum
is shown
Fig. is2.
To reduce
the computational
cost, in The
the outside
one dimensional
model
of the in
surface
used,
similar
to
the
soft
cube
model,
but
with
added
features
of
the
same
heat
bath
method
described
above, is
To reduce the computational cost, in the outside regions only a one dimensional model of the surface
so
that
both
surface
areas
are
kept
at
the
preseted
temperature.
These
particles
are
only
allowed
to
move
used,in similar
to theperpendicular
soft cube model,
butsurface.
with added
features
of athecalculated
same heat
bath method
described
above,
the
direction
to
the
Figure
3
shows
potential
energy
surface
(PES)
so that
both
surface
areas
are
kept
at
the
preseted
temperature.
These
particles
are
only
allowed
to
move
between
a platinum
surface andtoone
nitrogen
atom
(of a 3molecule)
for
an energy potential
corresponding
to the
potential
in the
direction
perpendicular
the
surface.
Figure
shows
a
calculated
energy
surface
(PES)
well depth of the interaction potential cpt-N and Fig. 4 shows the PES of CC-N for the graphite - nitrogen
between
a platinum surface and one nitrogen atom (of a molecule) for an energy corresponding to the potential
system: No differences can be seen between ID and 3D model. Furthermore it can be seen, that the PES
wellfor
depth
of the interaction potential and Fig. 4 shows the PES of for the graphite - nitrogen
platinum - nitrogen is very rough with deep "valleys" between the platinum atoms whereas the carbon
system:
No
dierences
can
be
seen
between
1D
andID3Dmodel
model.
it canofbean seen,
thatlayer
the isPES
is very smooth. To test the properties of the
the Furthermore
energy distribution
adsorbed
for surface
platinum
nitrogen
is
very
rough
with
deep
"valleys"
between
the
platinum
atoms
whereas
the
calculated (T f = 2QQK,T
= 500K). Figure 5 shows the number density, the rotational energy and carbon
the
surface
is
very
smooth.
To
test
the
properties
of the
modeldifferences
the energy
distribution
adsorbed
layer
is
kinetic energy of the adsorbed layer:
There are
only1D
minor
between
the ID of
andan3D
model area.
calculated
(
T
=
200
K
;
T
=
500
K
).
Figure
5
shows
the
number
density,
the
rotational
energy
and
the
The number density shows two adsorbed layers; (the first one one the first gridline and the second one on the
kinetic
the the
adsorbed
There
only minor
the 1D
andID3Dthan
model
area.
thirdenergy
gridlineoffrom
surface.layer:
Kinetic
and are
rotational
energydierences
show little between
higher values
in the
in the
The3Dnumber
density
shows
two
adsorbed
layers;
(the
rst
one
one
the
rst
gridline
and
the
second
one
on
the
area, i.e. the heat transfer in the ID region is slower.
third At
gridline
from
the
surface.
Kinetic
and
rotational
energy
show
little
higher
values
in
the
1D
than
in
the moment only pores of (nearly) circular shape are investigated, but any shape can be used easily, the
3D because
area, i.e.thetheconfiguration
heat transferis infilled
the step
1D region
slower.
by stepis by
searching for the next neighbours of existing molecules.
the parameters
heat bath
are calculated
described
above.
species
AtAfterwards
the moment
only poresofofthe(nearly)
circular
shape areautomatically
investigated,asbut
any shape
canAsbegasused
easily,
severalthe
monatomic
gases is(He,Ar,Kr)
andstep
diatomic
species (N^^O^)
The of
diatomic
because
conguration
lled step by
by searching
for the were
next chosen.
neighbours
existingmolecules
molecules.
were treated
as rigid rotators
quaternions
used for
describing their
rotational motion
to
avoid
the
Afterwards
theofparameters
of theand
heat
bath
are were
calculated
automatically
asaredescribed
above.
Asaccording
gas species
singularity
the
Euler
angles
[1].
In
the
source
and
sink
region
molecules
inserted
randomly
several monatomic gases (He; Ar; Kr) and diatomic species (N2; O2 ) were chosen. The diatomic molecules
were treated as rigid rotators and quaternions were used for describing their rotational motion to avoid the
singularity of the Euler angles [1]. In the source and sink region molecules are inserted randomly according
to the equilibrium distribution function of the preseted temperature and pressure up- and downstream of
Pt
sur
surf
N
C
gas
gas
533
N
PES between platinum and nitrogen FIGURE 4. Like Fig.3, but graphite - nitrogen
atom (Comparison of 3D model (inside the line) and
FIGURE
3. 3.PES
4. Like Fig.3, but graphite - nitrogen
1D
model (outside)
FIGURE
PESbetween
betweenplatinum
platinumand
and nitrogen
nitrogen FIGURE
FIGURE 4. Like Fig.3, but graphite - nitrogen
atom
(Comparison
of
3D
model
(inside
the
line)
and
atom (Comparison of 3D model (inside the line) and
ID 1D
model
(outside)
model
(outside)
FIGURE 3.
b
bR
R
R
10A
10A
1D
5 05
3
1D
568
568
10A
10A
10A
4
44
43
5 05
327327
299299
64
243
1D
1D
2
258 58
271 271
1D
1D
3D
3D
243
3D
3D
3D
0 40
0
3D
R
40
R
X
226
aR
c
c
X
X
343 3
364 3 43
X
X
226
a
X
10A
Adsorbate layer in front of Surface (Tsurf = 200K ; Tgas = 500K : a) Number density, b) rotational
FIGURE
Adsorbate
layer
frontofofSurface
Surface (T
(Tsurf
200K ; Tgas
K : a)
FIGURE
Adsorbate
layer
front
= 500
500K:
a) Number
Numberdensity,
density,b)b)rotational
rotational
surf == 200K;
gas =
energy/
kB , 5.
c) 5.
kinetic
energy/
kB inin
energy/kB
c) kinetic
energy/kB
energy//^,
c), kinetic
energy/&B
FIGURE 5.
the the
membrane
respectively.
Furthermore
molecules
are
from
the simulation,
simulation,after
aftercrossing
crossingthetheouter
outer
membrane
respectively.
Furthermore
molecules
are removed
removed
fromboundary
the
border
of
the
sink,
while
for
the
side
walls
of
the
source
a
periodic
and
for
the
backside
a
specular
to the
equilibrium
distribution
function
of
the
preseted
temperature
and
pressure
upand
downstream
of
border
therespectively.
sink, Usually
while forFurthermore
thedistance
side walls
ofthethesource
source
athe
periodic
boundary
and
for the
backside
a specular
reection
isofchoosen.
the
of
to
surface
is
about
20
times
the
mean
free
path.
In
the reection
membrane
molecules
are
removed
from
the
simulation,
after
crossing
the
outer
is choosen.
Usually
theside
distance
of longer,
the
source
toa the
surfaceboundary
aboutabout
20andtimes
the mean
free
In
the
downstream
region
the
mean
free
path
isof
soso the
distance
isisonly
only
10times
times
thepath,
path,apath.
because
border
of the sink,
while
for
the
walls
the
source
periodic
for
the backside
specular
the
downstream
region
the
mean
free
path
is
longer,
the
distance
is
about
10
the
because
else
thethe
simulation
region
very
reflection
issimulation
choosen.
Usually
the become
distance
of
source to the surface is about 20 times the mean free path. In
else
regionwould
would
become
verythelarge.
large.
the
time
integration
either
an
embedded
Runge-Kutta
(RK)
Method
of 55 order
order
withthe
stepsize-control
theFor
downstream
region
the
mean
free
path
is
longer,
so the distance
is only of
about
10
times
path, because
the time integration either an embedded Runge-Kutta
(RK) Method
with
stepsize-control
[3] or[3]For
aorLeap-Frog
(LF)
therotational
rotational
motion[1].[1].TheThe
a Leap-Frog
(LF)method
methodis isused,
used,which
whichisismodied
modied for
for the
the integration
integration ofof the
motion
th
th
534
else the simulation region would become very large.
For the time integration either an embedded Runge-Kutta (RK) Method of 5th order with stepsize-control
[3] or a Leap-Frog (LF) method is used, which is modified for the integration of the rotational motion [1]. The
speed of both integrations is approximately the same for the same precision (energy conservation); the RK
method needs six force evaluations per step, but this is compensated by a larger step size. The main drawback
of the RK method is the memory requirements for the intermediate steps, so that for larger systems only the
LF method can be used. The carbon lattice requires a much smaller stepsize, because the frequency of the
lattice vibrations is much higher, which can already be seen by the Debye Temperature which is 10 times
higher for carbon than for platinum (upt = 240K,o;c = 2230K). The indices of interacting atoms are stored
in a neighbour-list, which is updated from time to time, i.e. every 50 steps for LF and about every 10 steps
for RK integration. Afterwards this list is sorted and the interactions are grouped into vectors, so that inside
of one vector the force calculations, which are responsible for approx. 90% of the computational time, are
independent from each other and can be fully vectorized on the available Fujitsu Vector-Computer (NWT).
Typically about 80 vectors are created for the lattice, 250 vectors for interactions between gas and SD-surface
and 200 for interaction between gas and ID-surface, rather independent from the size of the simulation area,
but sligthly increasing with thick adsorbate layers.
Following sources for the potentials were used and the cross potentials, which are not explicitly mentioned,
are estimated by the Lorentz-Berthelot mixing rules: Pt — Pt, Pt — N2 [2]; C — C (harmonic lattice), N2 — C,
O2 - C [12]; O2 - O2 [6]; N2 - N2 [9]; He - He, Ar - AT, Kr-Kr,C-C (for gas-surface-interaction) [10].
Ill
SAMPLE CALCULATIONS
Figures 6-9 show the effusion of a gas mixture (helium/argon, 50/50, lOMPa) through a pore (diameter 55
A, length 43 A) of graphite or platinum into vacuum. Helium can neither be adsorbed by graphite or platinum,
while argon is trapped at the platinum surface, as can be seen by the number of argon atoms inside the pore.
The temperature of the whole system is either 500 K or 1000 K. First all simulations are performed until the
number of atoms inside the pore is constant, after that we start counting the molecules passing through the
pore. Figure 8 is one example of the beginning of a simulation and Fig. 9 the corresponding counting (for the
other cases the starting is not shown). The lines in the diagrams are for Knudsen diffusion calculated by Eq. 3
when the downstream pressure is negligible, although it is clear that the diameter and the length of such a small
pore cannot be determind ambiguously. Figure 6 shows a good agreement between simulation and expected
Knudsen flux (fiAr = 0.6S/timeunit(TU)l:fiHe — 2.1/TC7). In Fig. 7 the temperature is halved, i.e. the gas
density doubles and the Knudsen flux rates should increase by a factor of \/2. The density inside the pore is
roughly doubled for both species, but the fraction of argon is slightly higher than helium, i.e. argon seems
to be adsorbed at the wall. The flux of argon (fiAr — 1.1/TE7) increases even more than expected, while the
helium flux (fine — 2.4/TLQ is much lower than the Knudsen value. This might be caused by the adsorption
of argon or a stronger interaction between the species because of the increased density and reduced Knudsen
number. Figure 9 shows the same situation for a platinum pore: Here many argon atoms are trapped and the
flux of argon (fiAr = 1.6/TJ7) is much higher than in the graphite pore (Fig. 7) and higher than for Knudsen
flow, while the helium flux (hffe = 2.62/TC7) is slightly higher than in the graphite pore but still below the
Knudsen value. The flux rates are quite close to each other so that separation is reduced.
Figures 10 & 12 and Figs. 11 & 13 show similar simulations (dia. 50 A, length 43 A, 500K, lOMPa) for
nitrogen-argon (50/50) and nitrogen-helium (50/50) respectively. The masses of nitrogen and argon are not
too different, which results in similar Knudsen flux rates. Indeed in the graphite pore the simulation shows
quite similar flux rates (njv2 — 1.3/TC/, riAr = 1.1/TLQ, both a little bit higher than the expected values (Fig.
10). But in the platinum pore (Fig. 12), where argon is adsorbed much more than nitrogen, although nitrogen
can be trapped by platinum as was shown in Fig. 5, the argon flux (HAT — 1.6/TLQ is increased and becomes
even stronger than the nitrogen flux (fiN2 = 1.1/Tt/), i.e. the ratio is reverse compared to pure Knudsen
diffusion. Because of the large mass ratio, helium and nitrogen have very different Knudsen fluxes, but Fig. 11
shows, that the nitrogen flux (njv2 = 1.5/TJ7) is increased by the adsorption at the graphite pore walls, while
the helium flux (hffe = 2.6/TC7) is reduced compared to the Knudsen value, because the diameter of the pore
for helium is reduced by the adsorbed layer of nitrogen at the walls. The nitrogen flux (njv2 — 1.8/TJ7) in the
535
fc
Graphite - lattice 500 K
&
"Helium •
Argon
400
50
450 Time (nondim.)
55
°
° Time (nondim.)
FIGURE 7. Same like Fig. 6, but 500 K
FIGURE 6. Diffusion in pore (diameter 55 A,
length 43 A): Simulation (points) and Eq. 3 (lines);
He, Ar, graphite, 1000 K.
- Platinum - lattice 500 K
Helium •
Argon
250
200
150
100
50
0
50
100
150 200
250
300
Time (nondim.)
FIGURE 8. Beginning of the simulation of Fig. 9
0
300
350
400
450 Tmie (nondim.)
FIGURE 9. Same like Fig. 6, but Pt 500 K
platinum pore is even much stronger, because of the surface diffusion, whereas the helium flux (fine — 2.6/TC7)
is same as in the graphite pore.
IV
CONCLUSION
A molecular dynamics model of porous flow was developed: Two lattice materials (graphite, platinum) and
gas atoms as well as molecules with rotational degrees of freedom can be simulated. The model was used
to calculate the effusion of gas mixtures into vacuum. The results show a strong influence of the gas-surface
interaction on the flux rates, which do not follow the Knudsen flux rate when adsorption in the pore occurs. In
Knudsen flow lighter molecules travel faster than heavier ones, which gives the separation effect. At least for
the gas molecules investigated here, havier molecules have also stronger interaction potentials with the surface
and these components build adsorbate layers inside the pore and surface diffusion of this component occurs
which enlarges the flux for the heavier component so that the separation effect of the Knudsen flow is reduced
or even inversed. The presented simulations used very short pores and simulations with longer pores (220 A)
are going to be performed now.
The lattice is going to be extended to other materials and models. Silicon and graphite lattices can be
simulated by the Brenner-potential, which is a three body potential. Furthermore ionic crystals, which are
very important as membrane materials, are going to be implemented, but here special simulation techniques
are necessary, to handle the long range coulomb forces, i.e. Ewald summation or multipole methods. Ionic
crystals are made from different species which give rough and ordered structures on the surface [4].
536
100 ~50
^50
30
°
35
°
Time (nondim.)
100
1UU
50
°
Time (nondim.)
200
FIGURE 10. Same like Fig. 6, but N2, Ar, graphite, FIGURE 11. Same like Fig. 6, but N2, He,graphite,
500 K
500 K
400
700
5-H
<L>
400
350
300
150
250
200
100
150
100
50
0
550
FIGURE 12.
500 K
600
Time (nondim.)
Same like Fig. 6, but JV2,
0
350
40
55
°
Time (nondim.)
°
FIGURE 13. Same like Fig. 6, but N2,He,Pt,
500 K
A Knudsen compressor using the thermal transpiration effect in pores with a diameter of 20C)Awas investigated by Vargo and Muntz [11] experimentally and theoretically. It seems to be possible to extend the present
MD model to such pore diameters because all computations in the present work were done on one PC (PII
500-Dual) and two a -Workstations (Compaq XP1000) within only 3 weeks and the memory requirement was
about 60 MB. But the experimentally length ratios L/D > 100 cannot be treated by the MD method, because
the surface would require a number of atoms in the order of 106.
Recently experiments were published influencing the flow of molecules with internal degrees of freedom in
pores by magnetic fields by Hermans [4]. Levdansky [5] investigated the pumping of gases in pores by exciting
interal degrees of freedom by laserlight. Sone [8] reported the flux of a rarefied gas without average temperature
and pressure gradients. These effects are going to be included in the molecular dynamics model to further
investigate the influence of the internal degrees of freedom and the results are going to be compared with the
published experimental results.
ACKNOWLEDGEMENTS
Part of this work was done while staying at National Aerospace Laboratory as a postdoctoral fellow. Dr.
Koura is gratefully acknowledged for offering this stay and the Science Technology Agency of Japan is thanked
for the financal support.
537
REFERENCES
1.
2.
3.
4.
5.
6.
7.
Alien, M.P., Tildesley, D.J. Computer Simulation of Liquids, Clarendon, Oxford (1987)
Blomer, J., Beylich, A.E. Surface Sience 423 , pp. 127-133 (1999)
Cash, J.R., Karp, A.H. ACM Transactions on Mathematical Software 16, pp. 202-222 (1990)
Hermans, L.J.F. 19th Rarefied Gas Dynamics (Oxford), Oxford, Oxford Univ. Press, pp. 931-940 (1995)
Levdansky, V.V., 6th World Filtration Congress, (1993)
Kaltz, T.L., Long, L.N., Micci M.M. AIAA Paper 97-3332
Pham-Van-Diep, G., Keeley, P., Muntz, E.P., Weaver, D.P. 19th RGD (Oxford), Oxford, Oxford Univ. Press, pp.
713-721 (1995)
8.
9.
10.
11.
Sone, Y., Kato, K. Preprint , 1999 (see this proceedings, too)
Matsumoto, Y., Tokumasu T. 19th RGD (Oxford), Oxford, Oxford University Press, pp. 808-814 (1995)
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