JOURNAL OF CHEMICAL PHYSICS
VOLUME 110, NUMBER 8
22 FEBRUARY 1999
Fluid transport properties by equilibrium molecular dynamics.
II. Multicomponent systems
D. K. Dysthe,a) A. H. Fuchs, and B. Rousseau
Laboratoire de Chimie Physique des Matériaux Amorphes, Bâtiment 490, Université Paris-Sud,
91405 Orsay Cedex, France
M. Durandeau
Total Exploration Production, CSTF, Domaine de Beauplan, Route de Versailles,
78470 Saint-Remy-Les-Chevreuse, France
~Received 31 August 1998; accepted 17 November 1998!
The Green-Kubo formalism for evaluating transport coefficients by molecular dynamics has been
applied to multicomponent mixtures of flexible, multicenter models of linear and branched alkanes
and nitrogen and helium in the gas phase and in the liquid phase. Simulation results on binary
systems are summarized and trends in prediction using simple but realistic molecular models are
shown. New simulation results of N2 –n-pentane agree with experiment with a maximum deviation
of 36%, the greatest error being for pure n-pentane. Methodological aspects of simulating
multicomponent systems with trace components are studied, varying the system size and molecular
interaction potentials. It is shown that mixtures are treated representatively even when only one to
two molecules of a species are present in the simulated system, unless there is an extreme degree of
self-association. It is demonstrated that molecular dynamics may predict quantitatively ~7% and
11% deviation! the viscosity of a seven component ‘‘synthetic’’ natural gas. © 1999 American
Institute of Physics. @S0021-9606~99!51008-8#
I. INTRODUCTION
Knowledge of transport properties of multicomponent
gases and oils is of great economical importance both in
reservoir modeling, planning of transport and in design of
industrial plants. For mixtures of constituents of dissimilar
size, shape and polarity, traditional prediction methods for
viscosity and thermal conductivity need experimental data to
fit mixing rules,1 while for diffusion the few existing prediction methods deviate by up to almost an order of magnitude.2
The presence of trace amounts of components that either
have very different molecular weight or specific interactions
is known to modify the transport coefficients of the mixture.
This is one of the reasons for the difficulty of predicting the
transport coefficients of such mixtures. One possible route to
obtaining better prediction is to use molecular dynamics
~MD! that can predict thermophysical properties from models of molecular interactions only.
For more than 25 years molecular dynamics has been
used to study fluid transport properties. Such MD studies
have followed diverse trends:
~a!
~b!
~c!
~d!
The bulk of research articles in this field has been devoted to methodological development either on a model independent form or directed towards allowing the use of new
classes of molecular models. Today there are several computational techniques and classes of models that may be considered ‘‘standard’’ and reliable from a methodological
viewpoint. In the case of viscosity, both the Green-Kubo
~GK! formalism and the SLLOD algorithm can be used with
multicenter molecular models to yield the zero strain shear
viscosity. The methods have been shown to be in agreement
with each other22,23 and are, in principle, ready to be applied
to ‘‘realistic’’ models of real multicomponent natural gases
and oils. There are, however, still methodological problems
to resolve, notably the representativity of a multicomponent
mixture of a small total number of molecules. This work is,
to our knowledge, the first attempt to use MD to calculate
transport coefficients of realistic natural gas ~multicomponent! mixtures.
The development of new computational techniques like
the synthetic3 and direct4 ~boundary driven! nonequilibrium molecular dynamic ~NEMD! techniques.
The study of microscopic transport mechanisms; for
instance, the early studies of details of the velocity autocorrelation function,5 the rheological properties of
a!
Present address: Department of Physics, University of Oslo, P.O. Box 1048
Blindern, N-0316 Oslo, Norway; electronic mail: d.k.dysthe@fys.uio.no
0021-9606/99/110(8)/4060/8/$15.00
liquids under shear6 and the partitioning of fluxes in
fluid mixtures in a temperature gradient or in a compositional gradient.4
The development of new models of pair interactions
from the hard sphere and Lennard-Jones ~LJ! fluids to
ab initio models of atoms7 and multicenter molecular
models.8–16
The study of transport properties of certain fluids and
classes of fluids. Simulations to obtain fluid transport
properties at a wide range of states have been performed on the hard sphere fluid,17 the Lennard-Jones
fluid18 and lately on multisite models of alkanes,19,20
their mixtures and mixed with CO2 and N2 . 21
4060
© 1999 American Institute of Physics
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Dysthe et al.
J. Chem. Phys., Vol. 110, No. 8, 22 February 1999
A. Evaluation of MD as a prediction tool
In order to evaluate the usefulness of MD for prediction
of transport coefficients of real, natural gases we proceed in
four steps.
1. Mixtures of light and heavy alkanes
Molecular potential models of alkanes are all adjusted
for pure fluids. The cross interactions between small and
large alkanes are not necessarily well taken into account. We
have already performed a study of the steric shape and size
effects in two such mixtures21 and we will use the conclusions from this study to give a complete picture.
II. MOLECULAR DYNAMICS METHOD AND
POTENTIAL MODELS
The molecular dynamics ~MD! calculation of transport
coefficients by the Green-Kubo method and our implementation have been explained thoroughly in previous
articles.21,23 We calculate the viscosity, h, the thermal conductivity, l and the intradiffusion coefficients, D a
h5
l5
2. N2 , CO2 – alkane mixtures
Nitrogen and carbon dioxide are very often present in
natural gases. Whereas most alkanes ~except for ethane! have
virtually no quadrupolar interactions, the thermodynamic
properties of N2 and CO2 are known to be described properly
only for potential models taking quadrupolar interactions
into account. We want to keep the simpleness and speed of
computation of the united atom ~UA! models, but there exists no theory for how to combine these models with other
molecules which are modeled using quadrupolar interactions.
We have therefore chosen to use simple, two-site LJ models
of N2 and CO2 that have been adjusted to thermodynamic
data at a wide range of states. It is therefore necessary to
evaluate the effects of this simplification on the transport
properties of mixtures.
3. Specific effects of trace components
Natural gases usually have a few components that are
present in great numbers and some components like H2 S,
aromats and heavy hydrocarbons that are present in trace
amounts. The transport coefficients of a mixture may be
greatly affected by the presence of trace amounts of components with very different molecular weight or with specific
interactions.1 MD simulations use small systems with the
number of molecules being between 100 and 1000 to achieve
reasonable statistics in a few days computation on a good
workstation. The basic question we need to address is: in a
mixture with trace amounts of molecules with specific ~like
or cross! interactions, how small a system may one use and
still get a proper description of the transport mechanisms?
We have constructed four simple test cases, two two component and two three component atomic systems using LJ/s
model interactions for all atoms, but with different sizes,
well depths and mixing rules.
4. A realistic test case
In order to test the prediction of viscosity by MD for
realistic systems, we have chosen a seven component system
which is a synthetic, model natural gas with the components
methane, ethane, propane, nitrogen, n-butane iso-butane and
helium 4. It is synthetic in the sense that it has been mixed in
a lab with exact mole fractions and it is a model system in
the sense that it contains typical amounts of components
found in natural gases. We have chosen two state points, one
in the gas phase and one in the liquid phase.
4061
V
10k B T
E
V
E
3k B T 2
D a5
1
3N a
`
0
`
0
E
dt ^ P0S ~ t ! :P0S ~ 0 ! & ,
~1!
dt ^ Jq ~ t ! •Jq ~ 0 ! & ,
~2!
`
0
dt
( ^ vi~ t ! •vi~ 0 ! & .
iPa
~3!
Here V is the system volume, T the temperature, k B the Boltzmann factor, P0S the symmetric traceless pressure tensor, Jq
the heat flux, vi the instantaneous velocity of the center of
mass of molecule i, N a the number of molecules of type a
and t the time. We apply the molecular definition of P0S and
Jq . 23 We have not attempted to correct the thermal conductivity for the lack of partial molar enthalpies in the microscopic heat flux definition but in all cases where we have
compared the thermal conductivity from GK and NEMD we
have found that they agreed within precision.24
In an n-component system there are n(n21)/2 independent interdiffusion coefficients D i j . These may be defined in
multiple ways and in different reference frames. The literature contains more definitions of combinations of the coefficients for multicomponent systems than experimental data
and simulation data does not exist. The formal Green-Kubo
theory for D i j in multicomponent mixtures has been presented by Zhou and Miller.25 From a simulation point of
view, the simplest approach is to generalize the two component expression of the kinetic part of the interdiffusion coefficient directly
D Ki j 52
V2
3Nw i w j m i m j
E
`
0
dt ^ Ji ~ t ! •J j ~ 0 ! & ,
~4!
where Ji is the mass flux of component i, w i the mass fraction, N the total number of molecules, and m i the molecular
mass. If one calculates D Ki j for all j.i one has n(n21)/2
independent coefficients. All other definitions of kinetic interdiffusion coefficients may be obtained by linear combinations of these D Ki j . In order to obtain experimentally observable interdiffusion coefficients, one also needs the partial
molar volumes and mole fraction derivatives of the chemical
potentials. The advantage of using the kinetic part of the
interdiffusion coefficients ~which are directly obtainable by
GK! is that approximative theories26 connecting intradiffusion and interdiffusion use the D Ki j .
In Ref. 23 we showed that when proper care is taken to
assure equilibration and statistical independence of subaverages the method is reliable from gases to high density liquids. Molecular potential models found in the literature are
grossly inaccurate with respect to experiment close to the
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4062
Dysthe et al.
J. Chem. Phys., Vol. 110, No. 8, 22 February 1999
TABLE I. Potential parameters.
4
He
N2
C2 H6
CH4
s ii
~Å!
e ii /k B
~K!
2.58
3.2973
3.52
3.7327
10.223
36.32013
137.5
149.92
mi
~g mol21)
I
~Å2 g mol21!
d bond
~Å!
8.2477
26.933
1.085471
2.345
4
28
30
16
Ref.
d
f
b
e
d AUA @Å#
i-butane
3.77
98.1
15
CH3
CH
4.1
12
13
Bending potential, u u /k B 5(62500/2)(cos u2cos 1.9897) 2 K
0.0
0.0
1.54
1.54
g
Propane, n-butane, n-pentane, n-decanea AUA„3…
CH3
3.516
79.87
14
CH2
3.516
119.8
15
Bending potential, u u /k B 5(62543/2)(cos u2cos 1.9775) 2 K
0.40
0.18
1.533~1.545a!
1.533~1.545a!
n-pentane OPLS
CH3
3.905
CH2
3.905
Bond angles constrained
0.0
0.0
1.53
1.53
h
c
59.38
88.07
14
15
Torsion potentials
u t /k B 5 ( i50 a i cosi x K
a0
a1
a2
a3
a4
a5
a6
a7
a8
OPLS
1116.0 1462.0 21578.0
2368.0 3156.0 23788.0
AUA~3! 1001.35 2129.52 2303.06 23612.27 2226.71
1965.93 24489.34 21736.22 2817.37
a
For n-decane LJ/s site–site interaction and the longer bond lengths were used.
Reference 9.
c
Reference 10.
d
Reference 12.
e
Reference 13.
f
Reference 14.
g
Reference 15.
h
Reference 16.
b
solid-liquid transition,19,20 but none of the simulations reported here are at such high densities and low temperatures.
All simulations were performed in the canonical ~NVT!
ensemble using a molecular Nosé-Hoover thermostat. The
trajectories used for calculation of the transport properties
were 5 ns for the N2 –n-pentane mixtures with a time step
d t55 fs. The estimated precisions are 610% for the viscosities, thermal conductivities and interdiffusion coefficients
and 61% for the intradiffusion coefficients. For the two and
three component systems we used a time step of d t
• A« 11 /m 1 / s 1150.004 and trajectories of 23106 –23107
time steps ~greater number for the smaller systems!. For the
realistic, seven component mixture the trajectories were 10
ns and d t55 fs. The cutoff distances for the Lennard-Jones
intermolecular potentials were always 2.5 s i j , except for the
Lennard-Jones/spline ~LJ/s! potential that goes smoothly to
zero at about 1.74 s i j .
We have used a number of molecular potential models;
all the parameters of the realistic models are given in Table I.
For the alkanes we have used UA models taken from the
literature. The details of these models and how they are
treated by our MD program are given elsewhere.23 The potential parameters used for the three component LJ/s mixture
are given in Table II. The mixing rules used are s i j 5( s ii
1s jj)/2 and e i j 5 Ae ii e j j , unless otherwise specified in Table
II.
TABLE II. Two and three component system parameters. All components interact through single LJ/s poten3
tials. All simulations are performed at temperature k B T e 21
11 51.5 and density rs 1150.6.
Two component
xi
m i /m 1
i\ j
1
2
3
0.96875
0.03125
1.0
1.0
e i j / e 11
1
2
1.0
1.0
1.0
2.0
Three component
s i j / s 11
1
2
1.0
1.0
1.0
2.0
xi
m i /m 1
1
0.92593
0.00926
0.06482
1.0
2.0
0.5
e i j / e 11
2
3
1
s i j / s 11
2
3
1.0
0.5
1.837 1.0 0.85 1.15
0.5
0.1/1.5 0.6123 0.85 0.7 1.0
1.837 0.6123 1.5
1.15 1.0 1.3
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Dysthe et al.
J. Chem. Phys., Vol. 110, No. 8, 22 February 1999
4063
proximately as well for the mixture as for the pure components. If one for other reasons should want to improve the
cross interactions of the potential models, one should use
diffusion which is very sensitive to all cross interactions.
B. N2 , CO2 –alkane mixtures
FIG. 1. Deviation between MD prediction and experiment of transport coefficients. Legends A: squares: l of C2 H6 – C10 H22 , triangles: h of
C2 H6 – C10H22 circles: h of CH4 – C10H22 . Legends B: squares: D 12 of
CH4 – C10H22 , triangles left: D 1 of CH4 –C10 H22 , diamonds: D 2 of
CH4 – C10 H22 , triangles right: D 1 of C2 H6 – C10H22 , circles: D 2 of
C2 H6 – C10H22 Legends C: squares: h of CO2 – C2 H6 , triangles: h of
CO2 – C10 H22 at 311 K, diamonds: h of CO2 – C10H22 at 409 K. Legends D:
diamonds: h of N2 – C5 H12 ~OPLS! at 325 K, triangles: h of N2 – C5 H12
~OPLS! at 421 K, circles: h of N2 – C5 H12 @AUA~3!# at 325 K, squares: h of
N2 – C5 H12 @AUA~3!# at 421 K.
III. RESULTS AND DISCUSSION
A. Gas–n-decane mixtures
Figure 1 summarizes all systematic MD studies of the
transport coefficients h , l, D i and D 12 in binary mixtures ~at
least one component being an alkane! with realistic molecular models. The results from Ref. 21 and this study may be
summarized in some general trends. The viscosity of ethane–
n-decane and methane–n-decane ~triangles and circles in
subplot A! and the viscosity of CO2 –n-decane ~diamonds
and triangles in subplot C! show the same behavior with
mole fraction of the light component. The viscosity of pure
n-decane is underestimated, whereas that of the pure gas is
predicted within the precision of the simulations ~610%!.
The main trend is a correct, smooth interpolation between
the two pure fluid viscosities which are different by a factor
6–20 at constant temperature and pressure. There is a small
effect of an increasing deviation in the mixture. We have
found that viscosity and thermal conductivity are insensitive
to the attractive site–site interaction21 and we therefore attribute the increasing deviation in the mixture to steric shape/
size effects of the molecular cross interactions. The thermal
conductivity of ethane–n-decane ~squares in subplot A! always agrees with experiment within the precision. Subplot B
shows the deviations of the diffusion coefficients in methane/
ethane–n-decane. One observes that the intradiffusion coefficients are very sensitive to the cross interactions, the intradiffusion of the light component is too high at low
concentration of n-decane and the n-decane intradiffusion is
too low at low concentration of n-decane. The conclusion is
that the viscosity and thermal conductivity ~which are most
important in industrial applications! are not very sensitive to
cross interactions between the species and are predicted ap-
We showed in Ref. 21 that the viscosity and thermal
conductivity are not sensitive to the attractive part of the
site–site LJ cross interaction, whereas the diffusion coefficients are very sensitive. This gives some hope for predicting
h and l in mixtures containing molecules with dipolar or
strong multipolar interactions. We have already commented
on the CO2 –n-decane viscosities that show the same behavior as that of the alkane mixtures. The squares in subplot A
of Fig. 1 show that for CO2 –ethane the viscosity is always
within precision of the experimental results, irrespective of
the attractive site–site cross interactions. Another molecular
species that is often present in natural gases is N2 and it is
therefore important for our program to verify if the simple
two center LJ model type is sufficient for describing the
transport in N2 –alkane mixtures. There exists, to our knowledge, no viscosity data in the literature for such mixtures, so
in cooperation with Institut Français du Petrole27 we have
obtained some new data for the system N2 –n-pentane. The
state points are given in Table III together with our simulation results using two different n-pentane models, OPLS and
anisotropic united atom @AUA~3!# ~see Table I!. Because
there exists no density data for this mixture, we have compared the viscosities at equal pressures instead of equal densities as for the previous results. The deviation from the experimental viscosity is shown in Fig. 1, subplot D. The
results show the same general trends as for the gas–n-decane
systems. The difference between the OPLS ~optimized potentials for liquid simulation! and AUA~3! models is, however, a bit curious. It seems like the AUA~3! model treats the
cross interactions better than the OPLS model. There is a
qualitative difference between the models, the OPLS model
has a LJ interaction site centered on the carbon with a larger
diameter than the AUA~3! model which has the interaction
site displaced to the geometric center of the valence electrons
of the CHi group. One will need more data to confirm
whether there is a real difference in the effective cross interactions or whether it is an artifact due to poor statistics.
C. Specific effects of trace components
The transport coefficients of a mixture with a few ‘‘special’’ components may possibly be changed by ‘‘rare
events’’ which are not representatively sampled in a very
small system. We have constructed two two component and
two three component atomic systems to study the representativity of small systems with regard to the transport coefficients. We use LJ/s model interactions for all atoms, but with
different sizes, well depths and mixing rules. The potential
parameters and states are described in Table II. A number of
such interesting systems may be conceived, but our choice
was guided by the intention to study systems where the pres-
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4064
Dysthe et al.
J. Chem. Phys., Vol. 110, No. 8, 22 February 1999
TABLE III. Transport coefficients of N2 –n-pentane.
T
~K!
p
~MPa!
r
~g l21 )
~mPa s!
h
Dh
%
325.25
421.45
325.25
421.45
325.25
421.45
120.5
60.3
119.4
59.9
119.8
60.0
711.13
608.09
690.97
538.55
583.91
343.87
0.3160.02
0.1460.01
0.2060.01
0.07760.002
0.05460.002
0.031960.0007
236
232
222
230
23.6
27.2
0
325.25
120.7
697.30
0.3960.02
219
0
0.5
0.5
421.45
325.25
421.45
59.9
120.4
59.7
608.20
686.05
547.20
0.1760.01
0.1960.01
0.08560.002
217
226
223
x1
D1
~1029 m2 s21!
D2
~1029 m2 s21!
l
~W m21 K21!
3.8760.04
10.0260.05
6.0760.08
15.960.1
0.15460.008
0.10260.006
0.12660.004
0.08160.005
0.08160.003
0.05460.002
3.4460.05
0.13260.005
8.8360.07
5.660.1
14.2460.08
0.09360.002
0.12060.005
0.08060.002
AUA~3!
0
0
0.5
0.5
1
1
12.360.1
31.260.2
26.860.2
69.960.2
OPLS
ence of 1, 2 or 20 of a type of molecule ~keeping the number
fraction constant! might affect the transport properties differently.
11.360.2
28.360.2
lation function is effectively zero before the onset of the
effects of the periodic boundary conditions. This explains the
fact that the calculated viscosities show no dependence on
system size ~see Table IV!.
1. General trends
The most pronounced effect on the transport coefficients
upon changing system size is the increase in the intradiffusion D i shown in the left subplot of Fig. 2. We have plotted
D i as function of N 21/3 because the autocorrelation functions
are affected by correlations due to the periodic boundary
conditions and the characteristic time for the onset of such
perturbations scale with the length of the simulation cell. The
right subplot of Fig. 2 shows how the velocity autocorrelation functions are suppressed at earlier stages as the system
size decreases. It also shows that the shear stress autocorre-
2. Three component systems
All the D i (N) depend linearly on N 21/3 ~to within statistical precision! except D 2 (N) for the two component system @from now on denoted D 2,2com p (N)]. The slopes of
D pure(N), D 1,2comp(N), D 1,3comp(N) and D 3,3comp(N) depend
linearly on the respective diffusion coefficients extrapolated
to infinite system size ~axis intercepts of the linear fits displayed in Fig. 2!. The slope of D 2,3comp(N) deviates from this
TABLE IV. Transport coefficients of two and three component mixtures.
N
p
( e 11/ s 311)
( Ae 11m 1 / s 211 )
h
D1
( s 11Ae 11 /m 1 )
D2
( s 11Ae 11 /m 1 )
1
1
1
1
1
comp
comp
comp
comp
comp
32
108
500
1372
2048
1.43
1.46
0.85760.01
0.85360.008
0.18260.001
0.205160.0002
0.22260.002
0.22860.002
0.23360.002
2
2
2
2
2
comp
comp
comp
comp
comp
64
128
256
864
2048
1.44
1.43
1.46
1.48
1.47
0.86260.01
0.86160.01
0.86060.01
0.85960.01
0.85860.01
0.19760.001
0.207460.0008
0.214760.0005
0.22660.001
0.23060.002
3
3
3
3
3
3
3
comp
comp
comp
comp,
comp,
comp,
comp,
32
64
108
864
864
2160
2160
1.40
1.44
1.44
1.48
1.48
1.47
1.48
1.00560.02
1.00260.02
1.00360.008
1.0160.02
1.0060.02
1.0160.04
1.0760.06
0.157860.0005
0.171660.0008
0.179360.0002
0.197760.0005
0.197460.0005
0.203060.0005
0.202460.0005
0.27560.006
0.30060.005
0.2960.01
0.30760.004
0.30860.005
128
1024
2048
1.45
1.46
1.46
1.0760.04
1.0560.09
1.0760.1
0.116260.0005
0.127560.0004
0.130260.0003
0.035560.0008
0.047060.0009
0.050460.0009
C 1 -C 10
C 1 -C 10
C 1 -C 10
e 2250.1
e 2251.5
e 2250.1
e 2251.5
D3
( s 11Ae 11 /m 1 )
l
(k B Ae 11 /m 1 / s 211 )
3.0460.1
3.3660.02
0.19760.04
0.19360.007
0.19760.003
0.20460.002
0.20660.004
3.2660.07
3.4460.07
3.4760.07
3.5160.08
3.5460.07
0.10960.003
0.10160.002
0.11760.001
0.13360.001
0.11860.001
0.13660.002
0.13660.001
3.6260.07
3.8060.07
3.9260.07
4.260.2
3.9460.07
4.260.1
4.2360.07
2.7860.03
2.7860.07
2.8960.1
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J. Chem. Phys., Vol. 110, No. 8, 22 February 1999
Dysthe et al.
4065
FIG. 2. System size dependence of intradiffusion coefficients. Legends in
left subplot: circles: D of pure LJ/s fluid, squares: D 1,3 comp , triangles up:
D 2,3 comp, e 2250.1 , open triangles down: D 2,3 comp, e 2251.5 , diamonds: D 3,3 com p ,
triangles left: D 1,2 comp , filled triangles down: D 2,2 comp , filled circles:
D methane . Right subplot from left to right: normalized shear stress autocorrelation function for N5108, normalized velocity autocorrelation functions
for N532, 108, 864 and 2160.
linearity, but because of the low statistical precision one cannot decide whether this is some special ‘‘tracer effect’’ or
not.
Component 3 in the three component system has a
strong preferential interaction with component 1. The radial
distribution function g 13 has a very large first peak and g 33
has a small first peak and a large second peak which is at a
distance s 11 from its first peak ~see subplots B and C in Fig.
3!. The presence of this component changes the viscosity of
the system by 16%, the thermal conductivity by 18% and the
diffusion coefficient of component 1 by 213% relative to the
pure system values, even when only two particles of type 3
are present ~as is the case for N532!. The ratio of transport
coefficients of the three component system to the pure system is constant irrespective of the system size. We conclude
that for trace components associating with unlike components, even a small system with two trace particles is sufficient to represent the transport properties in the thermodynamic limit not only qualitatively, but quantitatively when
the pure system corrections in N 21/3 are applied.
One observes that changing the well depth e 22,3comp from
0.1 to 1.5 does not change the diffusion coefficient significantly, although the radial distribution function g 22,3comp
shows a much larger first peak in the latter case ~see subplot
B in Fig. 3!. It is clear that at this low concentration
(x 2 50.00926! the diffusion is completely dominated by
cross interactions and the greatly increased tendency for like
particles of type 2 to stick together does not slow down the
diffusional motion. One also observes that the presence of
component 2 ~for N>108) has no significant effect on the
other transport properties.
Since the three component system contains two species
in trace amounts, the interdiffusion coefficients28 D K1250.33
60.01, D K1350.2060.03 and D K2350.01860.009 are not of
much interest to test intradiffusion combination rules. D K12 is
dominated by D 2 , D K13 by D 3 and D K23 is hardly significant
since the fluctuations of the fluxes of the two tracer components are hardly correlated at all.
FIG. 3. Radial distribution functions of two and three component mixtures.
Subplot A, two component system: solid line: g 11(r), dashed lines: g 12(r),
long dashed line: g 22(r). Subplot B, three component system: solid line:
g 11(r), dashed line: g 22(r), e 2250.1, long dashed line: g 22(r), e 2251.5,
dot-dashed line: g 33(r). Subplot C, three component system: solid line:
g 12(r), long dashed line: g 13(r), dot-dashed line: g 23(r).
3. Two component ‘‘atomic’’ system
The two component system with atomic constituents is
constructed to be a pure system when one single particle of
species 2 is present and to have an extreme self-association
of species 2 ~see Table II!.29 The resulting radial distribution
functions for the 2048 particle system demonstrate this behavior ~see subplots A in Fig. 3!. Figure 2 displays the behavior of D 2,2comp(N) which displays one behavior at small
system sizes and a crossover to a lower diffusion coefficient
at larger system sizes. It is clear that two particles of type 2
are not sufficient to account for the possible situations occurring that contribute to the slowing down of the diffusional
motion. It is not easy to decide when D 2,2comp(N) reaches the
large system asymptotic behavior. One may well draw a
straight line through all the points within the error bars, but
then one must explain why the slope is so different from the
other slopes. The velocity autocorrelation function ~VACF!
of component 2 has the same decay rate as that of component
1 for N52048, i.e., the normalized VACF of component 2
fluctuates about the VACF of component 1 at large times.
We therefore expect the VACFs of the two components to be
subject to the same suppression due to system size, i.e., to
have more or less the same limiting slopes. If one expects the
slope to be similar to the surrounding slopes ~that change
linearly with axis intercept! only the largest system~s! containing 30–60 particles of component 2 are representative of
the infinite N fluid.
Downloaded 15 Jan 2002 to 129.240.85.171. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp
4066
TABLE V. Transport coefficients of a synthetic natural gas.
l
~g l21 )
~K!
~MPa!
~MPa!
~mPa s!
~mPa s!
~W m21 K21 )
Component
xi
Di
Di
0.731
0.061
0.034
0.158
0.006
0.002
0.008
(1029 m2 s21 )
97265
729612
560610
961610
473620
449620
28406100
(1029 m2 s21 )
38.860.1
30.960.3
27.160.4
39.360.3
23.660.8
2361
95.960.9
r
T
p MD
p exp
h MD
h exp
CH4
C2 H6
C3 H8
N2
n-butane
i-butane
He4
Dysthe et al.
J. Chem. Phys., Vol. 110, No. 8, 22 February 1999
19.7
338.9
2.69
2.36
0.014660.0005
0.01367
0.02460.001
317.8
301.2
44.4
41.53
0.03860.002
0.03432
0.07860.002
4. N-decane as tracer in methane
Real natural gases often contain trace amounts of heavy
hydrocarbons. We have therefore performed simulations to
probe the representativity of small systems with trace
amounts of n-decane in methane. The potential parameters
AUA~3! are given in Table I. For this system we used the
LJ/s potential for the site–site interactions to speed up the
computation and the same temperature T e 21
11 51.5 and density rs 31150.6 as for the other simplified model studies. The
mole fraction of n-decane was 1/128. The results ~see Table
IV! show that adding less than 1% of n-decane hardly
changes the pressure, but the viscosity is increased with
23%, the thermal conductivity is decreased by 19% and the
diffusion coefficient of methane is decreased by 44%. These
changes in transport coefficients from the pure to the
n-decane added system do not change with system size.
Thus, apart from the slight difference in slope between
D methane and D 3,3comp ~see Fig. 2!, there is no indication that
the N5128 system with a single n-decane molecule is not
representative for a system with the same mole fraction in
the thermodynamic limit.
D. Realistic test case
We have shown that a single molecule of a species is
sufficient to obtain a representative system for sampling the
transport coefficients unless this tracer species has an extreme self-associating tendency. We therefore proceed to apply the method to a realistic system. The mole fractions of
the seven component synthetic natural gas for which there
exists viscosity data30 are given in Table V together with the
temperatures and densities we have chosen to compare the
MD predictions with experiment. We have used 500 molecules in our simulations in order to have at least one molecule of the species with lowest mole fraction, iso-butane.
The results given in the same table show that the deviation is 7% for the gaseous state and 11% for the liquid state.
This agreement is as good as our earlier results for pure,
gaseous and liquid methane, ethane, CO2 , N2 and n-butane.
The previous conclusion for binary mixtures that mixture
predictions are approximately as good as the pure fluid pre-
dictions seems to be valid for multicomponent systems as
well. The added benefit of the equilibrium MD calculation of
the transport coefficients is that one obtains predictions for
all the transport coefficients in one simulation.
IV. CONCLUSION
The Green-Kubo formalism for evaluating transport coefficients by molecular dynamics has been applied to multicomponent mixtures of flexible, multicenter models of linear
and branched alkanes and nitrogen and helium in the gas
phase and in the liquid phase. We have presented new simulation results of N2 –n-pentane that agree with experiment
with a maximum deviation of 36%, the greatest error being
for pure n-pentane. We have summarized and compared all
transport coefficient results for binary mixtures with alkanes
as one component using realistic molecular models. The predictions of mixture transport coefficients are in general as
good as the predictions of the pure component properties.
We have studied several systems with one tracer component as function of system size N and different interaction
potentials. The diffusion coefficients increase linearly with
N 21/3. By comparing the behavior of the different systems
we show that mixtures are treated representatively even
when only one to two molecules of a species are present in
the simulated system unless there is an extreme degree of
self-association. In the latter case one may have to go to very
large systems (N;1000 and N trace;50) in order to represent
the dynamic behavior of the liquid. It is demonstrated that
molecular dynamics may predict quantitatively the viscosity
of a seven component synthetic natural gas in the gaseous
and liquid state.
ACKNOWLEDGMENTS
We would like to thank Total Exploration Production for
a grant for one of us ~D.K.D.!. We thank the Institut du
Dévelopement et des Ressources en Informatique Scientifique ~IDRIS! for a generous allocation of Cray T3E computer time.
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These are the values for the 2160 particle system. D K12 and D K13 have the
same system size dependence as the intradiffusion coefficients and D K23
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