JOURNAL OF CHEMICAL PHYSICS
VOLUME 112, NUMBER 17
1 MAY 2000
Fluid transport properties by equilibrium molecular dynamics. III.
Evaluation of united atom interaction potential models for pure alkanes
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
共Received 2 August 1999; accepted 9 February 2000兲
Results of new simulations for n-butane, n-decane, n-hexadecane, and 2-methylbutane at different
state points for seven different united atom interaction potential models are presented. The different
models are evaluated with respect to the criteria simplicity, transferability, property independence,
and state independence. Viscosities are increasingly underestimated 共up to 80%兲 and diffusion
coefficients are overestimated 共up to 250%兲 as the density increases and temperature decreases.
Clear evidence was found that the torsion potential is more important at high packing fractions and
for longer chains. The comparison of transport coefficients is argued to be a measure of ‘‘goodness’’
of the interaction potential models resulting in a ranking of the models. © 2000 American Institute
of Physics. 关S0021-9606共00兲51217-3兴
test seven different models.4 The deviation from experimental diffusion and viscosity data ranges from 1% to 30%. The
deviations for hexadecane reported are somewhat larger —
up to 63%5 and some of the state points are far outside the
range of experimental comparison.
Except for the two studies of n-decane at ambient condition, all MD studies of n-butane and n-decane have been
performed at moderately dense liquid states at temperatures
between the triple point and critical point. It has, however,
been demonstrated that viscosity and diffusion depend most
strongly upon the intermolecular parameters at high density
and low temperature.6 One of the main arguments for the
anisotropic united atom model 共AUA兲 versus the original
UA model is also based on the high density comportment of
the fluid.7
We therefore find it timely to evaluate the alkane models
that are most frequently used and most fitted to experimental
data. We have chosen to evaluate the models applied to one
short, one intermediate, and one long n-alkane and one
branched alkane at the state space extrema where transport
properties may be compared with experiment 共see Fig. 1兲.
In performing a study of this character one must choose
whether to compare simulation with experiment at equal
pressure or equal density. Thermodynamically one is free to
choose the state variables. From an engineering point of
view one normally wants prediction of transport coefficients
at a given pressure and temperature. Similarly, the experimentalist measures the temperature, T, and pressure, p, during the experiment, determination of the density, ␳ , requires
an additional experiment. For this reason there exist much
more transport data reported for the state variables T and p
than for the state variables T and ␳ , and the simpler choice
for our study would be to compare at equal T and p. Even so,
most systematic studies that compare simulation with experiment use equal T and ␳ . The reason is to be found in the
close theoretical relation between transport properties and
density. The exceptions are studies that use simulations to
I. INTRODUCTION
Molecular dynamics 共MD兲 is becoming an important
tool for probing molecular scale mechanisms that are not
readily available by experimental methods. With the increase
in computer speed it is also becoming a tool for quantitative
prediction of thermophysical properties. The basis for predictive use of MD is that interaction potential parameters are
adjusted to experimental data for some property and due to
an assumed property and state independence of the force
field one can apply it to predict other properties at other
states. The recent years has seen much refinement of ‘‘semirealistic,’’ multicenter interaction potentials. It is, however,
noteworthy that despite the increasing number of such proposed models, very few have been compared with experimental data for other properties than used in the parameter
adjustment at more than one or two state points. In the case
of transport properties the state points used are often repeated by each author with a new model without any reference to why this state point may be of interest. In fact, the
transport properties are often insensitive to details in the
models at the state points chosen.
Figure 1 shows the phase diagram and distribution in
temperature and density of transport property studies by MD
of n-butane, n-decane, and n-hexadecane using flexible, multisite molecular models. In the case of n-butane there have
been performed at least 14 transport coefficient studies using
five different variants of multicenter united atom 共UA兲 interaction potential models.1 Most of the studies have been performed at essentially the same state, all yielding transport
coefficients within 20% of experimental values. In the case
of n-decane we know of 15 studies2 that have been published
using five different models at essentially two different states.
In addition, Padilla and Toxvaerd3 chose two other states to
a兲
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
0021-9606/2000/112(17)/7581/10/$17.00
7581
© 2000 American Institute of Physics
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7582
J. Chem. Phys., Vol. 112, No. 17, 1 May 2000
Dysthe, Fuchs, and Rousseau
In Sec. II we will present a formal description and the
historical evolution of the molecular models, Sec. III gives a
brief description of the simulation details, and Sec. IV presents and discusses the results.
II. ALKANE MODELS FOR QUANTITATIVE
PREDICTION OF THERMOPHYSICAL PROPERTIES
FIG. 1. Vapor–liquid coexistence curves 共solid lines兲, distribution in
temperature–density space of experiments of viscosity 共inside dotted lines兲,
and diffusion 共inside long dashed lines兲 and state points for simulation by
other authors 共open squares兲 and this study 共closed triangles兲.
calculate ‘‘engineering numbers’’ for viscosity; like the
pressure–viscosity coefficient8 and the viscosity number.9
The aim of this study, however, is to present new results
that may be interpreted by theoretical concepts to improve
our understanding of the dependence of transport properties
on different parts of the molecular interaction potentials.
Whether these theoretical concepts be drawn from kinetic
theory of gases, reptation models of polymers, or transition
state theory, the common point is that transport properties
are more directly related to the density than to the pressure.
For example, the use of the Enskog theory as a basis of
understanding the mechanisms of transport properties in
dense fluids is based on the van der Waals picture of the
dense fluid as hard spheres moving in an attractive background field. According to the Enskog theory the transport
properties should be very sensitive to the repulsive part of
the potential model and the density, but not very sensitive to
the long-range attractive part. This is illustrated clearly in a
study by Cummings and Varner10 that showed that turning
off the electrostatic interactions in the simple point charge
共SPC兲 model of water changed the viscosity very little when
keeping the density constant, but that the pressure changed
by a factor of 10 or even changed sign.
When applying the van der Waals and Enskog approach
to realistic molecular liquids one must also consider the temperature dependence of the effective ‘‘hard sphere diameter.’’ In this way one may reduce the (T, ␳ ) dependence to
a dependence on a single reduced density; the density divided by a temperature-dependent close packed density that
contains the information of the temperature-dependent ‘‘hard
sphere diameter.’’ This reduced density or ‘‘packing fraction’’ is thus a single state variable of which the transport
coefficients are single-valued functions. The representation
of transport data as a function of such a packing fraction will
be shown to aid the interpretation of the results as it did in a
similar study by Allen and Rowley.11
It is desirable to describe molecular interactions by pair
interaction models that directly or indirectly reproduce and
predict experimental data of thermophysical properties with
a precision and calculation cost comparable or better than
experiment. We will adopt the main criteria of Allen and
Rowley11 for the ‘‘goodness’’ of interaction potential models: They should be 共1兲 simple in order to keep computing
time to a minimum, 共2兲 transferable in the sense that the
same group parameters can be used for all molecules of the
same family, 共3兲 property independent meaning that regressing the parameters for one property should give good predictions of other properties and 共4兲 state independent; the accuracy of prediction should not depend on the temperature,
density, and composition.
The simplest model fluid, the hard sphere fluid, has been
studied by kinetic theory and MD to form a basis for correlation and prediction of n-alkane transport properties.12 Soft
sphere models and Lennard-Jones 共LJ兲 have been used to
generalize and explain the connection between the correlation parameters and molecular parameters.13 Although much
of the basic physics of transport in dense fluids is arguably
contained in the van der Waals model, the predictive power
rests not on the model alone, but on additional empirical
parameters that must be regressed on every single property.
The LJ fluid has a phase diagram very similar to those of
alkanes. The transport properties of alkanes and the LJ fluid
also have the same general tendencies upon changing temperature and pressure. This simple, spherical model may be
used predictively by the simple application of the principle
of corresponding states. We will in this way use the LJ fluid
as a reference for the discussion of the results for multisite
interaction potential models.
A. United atoms models
Skipping rigid bodies, the next step on the alkane model
evolutionary chain appeared in 1975: the flexible, four center
n-butane model of Ryckaert and Bellemans.14 This type of
model collapses the CH, CH2 and CH3 groups into single
interaction sites, the so-called united atoms 共UA兲 described
by a soft pair interaction term. The distance between bonded
interaction sites is constrained, the angle between neighboring bonds is either constrained or is subject to a bending
potential, and the dihedral angle is subject to a torsion potential. Carbons separated by more than three bonds interact
via the same binary interaction potential as for intermolecular interactions to avoid an unphysical overlapping of sites.
The collapsed centers of force are characterized by the pair
interaction parameters of size, ␴ ii , and well depth, ␧ ii , the
mass, m, and the position of the center of mass and center of
force relative to the bonds, d AUA . In addition to ␴ ii and ␧ ii
one must specify the mixing rules, the cutoff distance, r c ,
and long-range corrections applied.
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J. Chem. Phys., Vol. 112, No. 17, 1 May 2000
Fluid transport properties by equilibrium molecular dynamics
7583
TABLE I. United atom interaction potential models.
Site–site potential
冉冉 冊 冉 冊 冊
␴ i j 12 ␴ i j 6
⫺
, r⭐r c ; u LJ⫽0, r⬎r c
r
r
1
r c ⫽2.5␴ i j,max , ␴ i j ⫽ 2 ( ␴ ii ⫹ ␴ j j ) and ␧ i j ⫽ 冑␧ ii ␧ j j
uLJ⫽4␧ i j
Site parameters and „constrained… bond lengths
OPLS
OPLS
OPLS
SKS
SKS
SMMK
SMMK
SMMK
AUA共2兲
AUA共2兲
AUA共3兲
AUA共3兲
CH
CH2
CH3
CH2
CH3
CH
CH2
CH3
CH2
CH3
CH2
CH3
OPLS
SKS, SMMK
AUA共2兲, AUA共3兲
␧ ii /k B
共K兲
3.850
3.905
3.905
3.93
3.93
4.1
3.93
3.77
3.527
3.527
3.516
3.516
40.26
59.38
88.07
47
114
12
47
98.1
80
120
79.87
119.8
mi
(g mol⫺1 )
13
14
15
14
15
13
14
15
14
15
14
15
d AUA
共Å兲
d bond
共Å兲
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.37
0.275
0.40
0.18
1.53
1.53
1.53
1.54
1.54
1.54
1.54
1.54
1.545,a1.533b
1.545,a1.533b
1.545,a1.533b
1.545,a1.533b
Bending potential
Constrained bending angle ␪ ⫽1.9548
u ␪ /k B ⫽(62 500/2)( ␪ ⫺ ␪ 0 ) 2 K, n-alkanes: ␪ 0 ⫽1.9897, branched: ␪ 0 ⫽1.9548
u ␪ /k B ⫽(62 543/2)(cos ␪⫺cos ␪0)2 K, ␪ 0 ⫽2.0001, a ␪ 0 ⫽1.9775b
a0
OPLS n-butane
OPLS n-alkanec
SMMK branched
AUA共2兲
AUA共3兲
␴ ii
共Å兲
Torsion potential
a1
1031.36
1009.99
428.7
1037.76
1001.35
a6
⫺4489.34
2037.82
2018.95
895.08
2426.07
2129.52
a7
⫺1736.22
a2
a3
158.52
136.37
223.7
81.64
⫺303.06
a8
2817.37
⫺3227.70
⫺3165.30
⫺1765.1
⫺3129.46
⫺3612.27
u t /k B ⫽ 兺 i⫽0 a i cosi ␹ K
a4
a5
⫺163.28
2226.71
⫺252.73
1965.93
a
Adjusted for n-decane and used for n-decane and n-hexadecane.
Adjusted for n-pentane, used for n-butane.
c
Used also for SKS and SMMK for all n-alkanes including n-butane.
b
We will give a general, formal description of the UA
models and shortly review the most important UA model
parameter sets published. The experimental and theoretical
basis for the different parameter sets is briefly presented as a
background for the discussion of their merits.
u d ⫽k d 共 d⫺d 0 兲 2 ,
B. General model description
The collapsed centers of force are characterized by the
pair interaction parameters ␴ ii and ⑀ ii , 15 the mass, m, and
the position of the center of mass and center of force relative
to the bonds. Different sets of proposed pair interaction parameters are given in Table I.
Toxvaerd noted that the hydrogens in the CHi groups
contribute to the united atom site potential and that the center
of the UA potential should be displaced relative to the carbon center by a distance d AUA :
rc f ,i ⫽ri ⫹d AUA•
兺 j ri ⫺r j
兩 兺 j ri ⫺r j 兩
⫽ri ⫹d AUAei ,
where ri is the carbon position, r j are the positions of the
carbon atoms bonded to ri , and ei is the unit vector from the
carbon to the center of force 共cf兲.
The intersite bonds have been modeled by some authors
by a harmonic potential u d around the equilibrium C – C distance d 0 ⫽ 兩 ri ⫺r j 兩 :
共1兲
共2兲
where k d is the bond spring constant. In this work we have
only used models with constrained bond length. One assumes that the bond vibrations have a much higher frequency
than other motions in the system and that this degree of
freedom does not couple significantly to the other degrees of
freedom. The constrained bond lengths are given in Table I.
The angle between two neighboring bonds, ␪ , may be
expressed by the positions of the three carbon atoms involved:
cos ␪ ⫽ 共 r j ⫺ri 兲 • 共 rk ⫺r j 兲
and the bending potential used in this work has three forms:
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7584
J. Chem. Phys., Vol. 112, No. 17, 1 May 2000
Dysthe, Fuchs, and Rousseau
FIG. 2. Molecular geometry. Left-hand side: Skeleton diagram of
2-methylpentane showing the carbon positions 共open circles兲, centers of
force 共closed circles兲, and bonds. Right-and side: Illustration of the two
dihedral angle definitions.
u ␪ ⫽constrained,
共3兲
k␪
u ␪⫽ 共 ␪ ⫺ ␪ 0 兲2,
2
共4兲
u ␪⫽
k␪
共 cos ␪ ⫺cos ␪ 0 兲 2 ,
2
共5兲
where ␪ 0 is the equilibrium angle and k ␪ is the bond spring
constant. The parameters ␪ 0 and k ␪ are given in Table I.
Between two carbons that are separated by three bonds
the interaction is modeled by a torsion potential that depends
on the dihedral angle ␹ :
cos ␹ ⫽⫺cos ␾ ⫽⫺
共 ri j ⫻r jk 兲 • 共 r jk ⫻rkl 兲
冑1⫺ 共 ri j •r jk 兲 2 冑1⫺ 共 r jk •rkl 兲 2
.
共6兲
It should be noted that both definitions ␹ and ␾ are used in
the literature, ␹ ⫽0 and ␾ ⫽ ␲ in the equilibrium 共trans兲 conformation of a normal alkane. The expressions of the torsion
potential u t is used in several forms:
u t⫽
兺 a i,1 cosi ␹ ,
i⫽0
共7兲
3
u t ⫽a 0,2⫹0.5
兺 a i,2关 1⫹ 共 ⫺1 兲 i⫹1 cos i ␾ 兴 .
i⫽0
共8兲
The Fourier form, Eq. 共8兲, is easily transformed into the
power form yielding the following relations between the coefficients:
a 0,1⫽a 0,2⫹0.5a 1,2⫹a 2,2⫹0.5a 3,2 ,
a 1,1⫽⫺0.5共 a 1,2⫺3a 3,2兲 ,
a 2,1⫽⫺a 2,2 ,
a 2,1⫽⫺2a 3,2 .
The parameters summarized in Table I have generally been
fitted to spectroscopic data and molecular mechanics potential minima and some have later been fitted to data on population of different conformations.
For branched alkanes we will distinguish between
two types of torsion potentials: X–CH2 –CH2 –Y and
X–CH–CH2 –Y, where X and Y may be any CHi group. In
Fig. 2 the torsion around the bond 2–3 is of the first type and
torsion around bond 3–4 of the second type. For computational convenience the second torsion is split into two separate contributions, u t,1 : 2–3–4–5 and u t,2 : 2–3–4–6 with
the ‘‘ideal’’ total torsion potential around 3–4 regained by
adding the two terms: u t ( ␹ )⫽u t,1( ␹ ⫺ ␪ 0 /2)⫹u t,2( ␹
⫹ ␪ 0 /2). For the calculations of the heat flux one should shift
the potential to assure that u t ( ␹ ⫽0)⫽0, which is not the
case for all published X–CH–CH2 –Y potential parameters.
Carbons separated by more than three bonds interact via
the same binary interaction potential as for intermolecular
interactions. This to avoid an unphysical overlapping of
sites. The equilibrium state of a molecule will thus have a
nonzero potential energy. The effect on the microscopic heat
flux is assumed to be negligible.
1. Ryckaert and Bellemans and variants
In a preliminary communication Ryckaert and Bellemans 共RB兲14 reported their simulations on a flexible, multicenter, united atoms molecular model of n-butane. The flexibility due to torsional motion 共bond lengths and bond angles
being fixed兲, was reported to cause a lack of backscattering
observed in the velocity autocorrelation function for atomic
liquids at comparable densities. This was the first reported
effect of going beyond atomistic models. The torsion potential was based on ‘‘somewhat controversial’’ experimental
data. This parameter set 共often referred to as RB1兲 and a
slightly adjusted version 共RB216兲 have been extensively used
for intercomparison of simulation methods. Since it was not
intended as a quantitative and accurate parameter set, we will
not include it or any of its truncated variants in our comparison.
2. UA optimal potential functions for liquid
simulations
The first extensive adjustment of parameters for a class
of molecules was done by Jorgensen, Madura, and
Swenson17 in 1984. Their optimized potential functions for
liquid simulations 共OPLS兲 for linear, branched, and cyclic
hydrocarbons were constructed to be transferable. The bond
lengths and angles are constrained and were based on microwave experimental data. The torsion potential was fitted to
molecular mechanics 共MM兲 calculations. Interaction sites
separated by more than three bonds interact with LJ potentials, but the potential parameters 共fitted to intramolecular
MM兲 are different from the intermolecular LJ parameters.18
The intermolecular potential parameters were optimized for
12 UA groups by fitting to density and heat of evaporation
data of 15 liquids at 1 atm and 25 °C. The final set had an
average deviation in density 共at ambient conditions兲 of 2.3%.
Although this seems like excellent agreement it should be
noted that a 2% change in density may change transport
coefficients by 30%.
3. Smit, Karaborni, and Siepman and variants
Smit, Karaborni, and Siepman19 found that in order to
reproduce the vapor liquid equilibria and critical points of
liquid n-alkanes 共n-pentane to n-hexadecane兲 they needed to
regress a new set of parameters. They used the torsion potential of the OPLS model and regressed the LJ site parameters keeping ␴ CH2 ⫽ ␴ CH3 . The most important parameter
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J. Chem. Phys., Vol. 112, No. 17, 1 May 2000
Fluid transport properties by equilibrium molecular dynamics
7585
TABLE II. n-butane transport coefficients.
␳
共kg m⫺3 )
␰
Experiment
AUA共2兲
AUA共3兲
SKS
SKS/2
SMMK
SMMK/2
OPLS
31.6
0.023
Experiment
AUA共2兲
AUA共3兲
SKS
SKS/2
SMMK
SMMK/2
OPLS
32.9
Experiment
AUA共2兲
AUA共3兲
SKS
SKS/2
SMMK
SMMK/2
OPLS
500.0
Experiment
AUA共2兲
AUA共3兲
SKS
SKS/2
SMMK
SMMK/2
OPLS
732.3
Model
Experiment
AUA共2兲
AUA共2兲
AUA共2兲
AUA共2兲
AUA共2兲
39.9
119.5
227.9
377.1
441.1
0.029
0.368
0.756
0.033
0.098
0.187
0.309
0.362
␩
␭
共W m⫺1 K⫺1 )
T
共K兲
p
共MPa兲
(10⫺3 Pa s兲
D
(10⫺9 m2 s⫺1 )
510.9
518.7
510.7
499.4
519.2
487.9
498.0
491.0
2.06
2.2⫾0.1
2.1⫾0.1
2.0⫾0.1
2.1⫾0.2
2.0⫾0.1
2.0⫾0.1
2.0⫾0.1
0.0130a
0.0150⫾0.001
0.0150⫾0.0008
0.0124⫾0.0009
0.0134⫾0.0009
0.0140⫾0.0008
0.0142⫾0.0007
0.0136⫾0.001
587⫾9
576⫾10
501⫾9
564⫾12
549⫾13
547⫾11
572⫾7
0.020⫾0.001
0.020⫾0.002
0.018⫾0.002
0.022⫾0.002
0.018⫾0.001
0.016⫾0.003
0.014⫾0.001
377.7
372.2
375.3
367.5
376.7
364.3
367.8
359.5
1.38
1.4⫾0.1
1.5⫾0.1
1.3⫾0.1
1.3⫾0.1
1.4⫾0.1
1.4⫾0.1
1.4⫾0.1
0.0101a
0.0101⫾0.0005
0.0108⫾0.0007
0.0095⫾0.0005
0.0096⫾0.0005
0.0101⫾0.0005
0.0098⫾0.0005
0.012⫾0.001
401⫾7
401⫾9
359⫾9
388⫾8
392⫾7
411⫾8
427⫾5
0.013⫾0.001
0.015⫾0.001
0.014⫾0.001
0.013⫾0.002
0.014⫾0.001
0.015⫾0.001
0.012⫾0.001
510.9
503.3
497.4
500.8
509.8
519.3
505.7
509.6
63.8
64⫾5
53⫾5
57⫾6
60⫾5
62⫾6
57⫾5
58⫾5
0.0976a
0.086⫾0.004
0.078⫾0.004
0.102⫾0.003
0.096⫾0.004
0.087⫾0.003
0.082⫾0.004
0.091⫾0.002
19.6⫾0.7
19.1⫾0.6
15.9⫾0.5
17.2⫾0.3
20.3⫾0.5
18.6⫾0.7
18.5⫾0.1
0.0724c
0.083⫾0.001
0.080⫾0.001
0.085⫾0.003
0.082⫾0.002
0.076⫾0.003
0.073⫾0.003
0.074⫾0.002
150
162.3
150.5
157.0
149.8
162.3
154.4
149.7
24.1
35⫾5
⫺36⫾6
141⫾5
124⫾6
45⫾8
31⫾7
71⫾6
1.77b
0.69⫾0.06
0.79⫾0.02
2.2⫾0.3
3⫾1
0.72⫾0.05
0.81⫾0.08
1.35⫾0.1
0.452c
0.93⫾0.02
1.21⫾0.08
0.32⫾0.02
0.27⫾0.02
0.91⫾0.06
0.84⫾0.04
0.50⫾0.01
0.181c
0.186⫾0.006
0.168⫾0.005
0.18⫾0.01
0.19⫾0.01
0.155⫾0.005
0.159⫾0.006
0.173⫾0.005
444
438.2
443.9
443.7
440.9
440.3
␩ expa
2.01⫾0.01
4.63⫾0.01
5.96⫾0.03
9.55⫾0.07
18.66⫾0.10
0.012⫾0.0005
0.015⫾0.0005
0.022⫾0.001
0.040⫾0.005
0.055⫾0.007
412⫾4
136⫾2
66.49⫾0.61
32.53⫾0.54
24.12⫾0.37
0.0118
0.0152
0.026
0.047
0.063
From Lee 共Ref. 28兲.
From Diller and Van Poolen 共Ref. 29兲.
c
From correlation of Assael et al. 共Ref. 12兲.
a
b
for the change in critical temperature with chain length was
␧ CH2 . This parameter set, called SKS, is displayed in Table I.
Siepmann et al.20 extended the SKS parameter set to
branched alkanes. To obtain satisfactory agreement with experimental vapor liquid equilibrium 共VLE兲 data of heptane
isomers they fitted new ␧ to CH3 groups on ethyl and methyl
side chains. In the same paper they proposed a new parameter set 共which we will call SMMK兲 with one unique parameter for all CH3 groups, but with ␴ CH2 ⫽ ␴ CH3 . The adjustment was also in this case done so as to fit the critical points.
After our simulations were started Martin and Siepmann21
readjusted the parameters of this model. The changes are so
small ( ␴ CH3 ⫽3.77→3.73 Å, ␧ CH3 /k B ⫽98.1→98 K, ␴ CH2
⫽3.93→3.95, Å and ␧ CH2 /k B ⫽47→46 K兲 that we do not
believe they will alter our main conclusions.
4. Anisotropic united atoms models
Toxvaerd7 found that UA models were not able to reproduce experimental pressure vs molar volume of n-alkanes
while keeping the ratio ␧ 11 /␧ 22⬇1.5. He proposed to introduce the anisotropy of the CHi groups while keeping the
simplicity of the UA approach. This was achieved by shifting the center of force of the UA a distance d AUA from the
carbon position to the geometrical center of the valence electrons of the CHi group. The new model, AUA共1兲 represented
a more correct equation of state for n-alkanes. Padilla and
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7586
J. Chem. Phys., Vol. 112, No. 17, 1 May 2000
Dysthe, Fuchs, and Rousseau
TABLE III. n-decane transport coefficients.
␳
共kg m⫺3 )
␰
Experiment
AUA共2兲
AUA共3兲
SKS
SKS/2
SMMK
SMMK/2
OPLS
559.9
Experiment
AUA共2兲
AUA共3兲
SKS
SKS/2
SMMK
SMMK/2
OPLS
␩
D
(10⫺9 m2 s⫺1 )
␭
共W m⫺1 K⫺1 )
T
共K兲
p
共MPa兲
(10⫺3 Pa s兲
0.470
510.9
502.3
512.2
519.8
501.4
521.5
509.5
499.2
5.5
9.8⫾3
13⫾4
3⫾4
⫺2⫾4
5⫾4
3⫾4
⫺22⫾3
0.144a
0.127⫾0.005
0.121⫾0.006
0.119⫾0.005
0.116⫾0.003
0.114⫾0.004
0.110⫾0.005
0.122⫾0.004
12.3d
8.8⫾0.2
8.9⫾0.3
9.9⫾0.4
9.8⫾0.2
10.5⫾0.2
11.5⫾0.4
12.0⫾0.2
0.071d
0.078⫾0.002
0.079⫾0.003
0.064⫾0.002
0.061⫾0.001
0.061⫾0.003
0.062⫾0.004
0.057⫾0.002
724.7
0.685
300
304.1
293.4
295.8
289.7
293.9
297.0
287.6
0.1
12⫾6
⫺2⫾5
⫺1⫾7
⫺8⫾5
⫺15⫾6
⫺10⫾6
⫺57⫾5
0.773b
0.51⫾0.03
0.48⫾0.02
0.61⫾0.05
0.48⫾0.04
0.45⫾0.03
0.42⫾0.03
0.51⫾0.03
1.48d
1.38⫾0.04
1.54⫾0.03
1.44⫾0.03
1.79⫾0.03
1.86⫾0.06
2.41⫾0.05
1.98⫾0.02
0.134d
0.137⫾0.002
0.135⫾0.003
0.113⫾0.003
0.110⫾0.003
0.103⫾0.003
0.104⫾0.003
0.103⫾0.004
Experiment
AUA共2兲
AUA共3兲
SKS
SKS/2
SMMK
SMMK/2
OPLS
783.6
0.744
293.2
293.7
292.6
294.6
288.3
297.1
292.9
284.1
100
103⫾6
93⫾7
104⫾8
93⫾6
70⫾7
68⫾6
17⫾6
2.44c
1.6⫾0.2
1.2⫾0.2
1.31⫾0.08
1.05⫾0.09
0.8⫾0.1
0.70⫾0.05
1.00⫾0.07
0.58d
0.56⫾0.02
0.66⫾0.02
0.64⫾0.02
0.84⫾0.02
0.97⫾0.03
1.37⫾0.04
0.88⫾0.02
0.168d
0.166⫾0.005
0.165⫾0.009
0.148⫾0.005
0.143⫾0.003
0.132⫾0.005
0.135⫾0.004
0.138⫾0.004
Experiment
AUA共2兲
AUA共3兲
SKS
SKS/2
SMMK
SMMK/2
OPLS
820.9
0.783
286
297.8
286.9
282.9
284.7
284.8
287.3
278.1
215⫾8
177⫾7
196⫾8
196⫾6
141⫾8
148⫾7
95⫾6
5.8d
3.2⫾0.3
3.5⫾0.6
2.9⫾0.2
2.2⫾1
1.7⫾0.2
1.0⫾0.1
2.8⫾0.3
0.24d
0.31⫾0.01
0.238⫾0.005
0.276⫾0.006
0.38⫾0.1
0.53⫾0.02
0.83⫾0.03
0.43⫾0.04
e
e
e
e
e
e
e
Model
From Lee 共Ref. 28兲.
From Gehrig and Lentz 共Ref. 30兲.
c
From Zhou 共Ref. 31兲.
d
From correlation of Assael et al. 共Ref. 12兲.
e
Too short initial relaxation time.
a
b
Toxvaerd3 later adjusted the torsion potential and the d AUA to
obtain better agreement with self diffusion coefficients22 of
n-pentane and n-decane. We will call this parameter set
AUA共2兲. Toxvaerd23 refined the parameter set 关which we
will call AUA共3兲兴 even further in fitting it to pressure data at
even higher temperatures and pressures.
properties simultaneously; nonequilibrium methods are always property specific. We calculate the viscosity, ␩ , the
thermal conductivity, ␭, the intradiffusion coefficients, D a ,
by the usual correlation function integrals:
␩⫽
V
10k B T
冕
V
冕
III. SIMULATION DETAILS AND DATA TREATMENT
The details of the MD programs used, the special methodological considerations at the extreme fluid states, and the
treatment of the simulation data to obtain error estimates
have been published previously.24 That publication also contains a discussion on the choice of the NVE vs the NVT
ensemble. We use the RATTLE25 algorithm to solve the constrained equations of motion. In order to study the property
independence of the models we have chosen to use the
Green–Kubo 共GK兲 formalism that yields all the transport
␭⫽
3k B T 2
D a⫽
1
3N a
⬁
0
⬁
0
冕
dt 具 P0S 共 t 兲 :P0S 共 0 兲 典 ,
dt 具 Jq 共 t 兲 •Jq 共 0 兲 典 ,
⬁
0
dt
兺 具 vi共 t 兲 •vi共 0 兲 典 .
i苸a
共9兲
共10兲
共11兲
Here V is the system volume, T the temperature, P0S the
symmetric traceless pressure tensor, Jq the heat flux, vi the
instantaneous velocity of the center of mass of molecule i, N
the total number of molecules, N a the number of molecules
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J. Chem. Phys., Vol. 112, No. 17, 1 May 2000
Fluid transport properties by equilibrium molecular dynamics
7587
TABLE IV. n-hexadecane transport coefficients.
T
共K兲
p
共MPa兲
(10⫺3 Pa s兲
␩
D
(10⫺9 m2 s⫺1 )
␭
共W m⫺1 K⫺1 )
0.453
563
554.1
556.5
1.422
65⫾86
⫺41⫾34
0.195a
0.169⫾0.009
0.153⫾0.006
7.97c
6.4⫾0.1
7.8⫾0.3
0.064c
0.080⫾0.003
0.062⫾0.002
0.694
348
352.0
339.8
0.1
148⫾54
⫺175⫾53
1.231b
0.83⫾0.06
0.60⫾0.05
0.774c
0.87⫾0.03
1.30⫾0.04
d
d
323
327.7
328.9
147.9
1622⫾70
1026⫾69
7.92b
4.2⫾0.5
2.0⫾0.5
0.168c
0.180⫾0.005
0.41⫾0.01
d
d
␳
共kg m⫺3 )
␰
Experiment
AUA共2兲
SMMK
574.8
Experiment
AUA共2兲
SMMK
735.9
Experiment
AUA共2兲
SMMK
824.1
Model
0.777
From Matthews et al. 共Ref. 32兲.
From Tanaka et al. 共Ref. 33兲.
c
From correlation of Assael et al. 共Ref. 12兲.
d
Too short initial relaxation time.
a
b
of type a, w i the mass fraction, m i the molecular mass, and t
the time. We apply the molecular definition24 of P0S and Jq .
All the simulations were performed with 108 molecules.
Each configuration was equilibrated and stabilized at the desired temperature during 1 ns. The production runs were performed for 8 ns in the NVE ensemble with 4 fs time steps.
During each run, ten subaverages were saved to disk as the
basis for statistical analysis. The loss of total energy per
nanosecond in an NVE run was never more than 0.3 times
the standard deviation of the kinetic energy. We have been
careful to verify that the results are converged and within the
stated error bars. This is especially important at high densities and low temperatures.24
We chose to perform the simulations in the microcanonical ensemble and not in the canonical to avoid any possible
interference between a thermostat and the correlation function calculations.24,26 This leads to deviations between desired temperature and actual temperature, measured in the
simulation. Before comparing the simulation results with the
experimental data we correct the results to the temperature of
the experiments in the following manner:
K̂ sim共 T exp兲 ⫽
K Assael共 T exp兲
K 共 T 兲 ⫽ ␣ K sim共 T sim兲 ,
K Assael共 T sim兲 sim sim
共12兲
where K苸 兵 ␩ ,D,␭ 其 , K̂ is the corrected transport coefficient,
and the subscripts sim, exp, and Assael signify simulation,
experimental, and from the Assael correlation, respectively.
The Assael correlation12 is based on the Enskog theory and a
careful evaluation of available transport data. It is known to
represent all reliable experimental data on viscosity, thermal
conductivity, and diffusion to experimental accuracy or better. As long as one does not exceed the region in state space
for which the correlation was fitted we estimate that the correction scheme introduces an additional error of no more
than 10␣ %, i.e., 1/10 of the correction itself.
IV. RESULTS AND EVALUATION OF MOLECULAR
MODELS
We will present the results of new simulations for
n-butane, n-decane, n-hexadecane, and 2-methylbutane at
different state points in order to evaluate the different UA
interaction potential models with respect to the criteria simplicity, transferability, property independence, and state independence. We will also compare these data to previously
published data for the LJ fluid27 and all atom 共AA兲11 interaction potential models. In Tables II–V we present the results of the simulations for seven different models at the state
points indicated by triangles in Fig. 1. The estimated accuracy reported in Tables II–V are standard deviations computed as described in a previous publication,24 but in general
the viscosities and thermal conductivities are accurate to
5%–10% and the diffusion coefficients are accurate to
1%–2%.
Two of the seven models, SKS/2 and SMMK/2, are exactly equal to SKS and SMMK, respectively, except that the
OPLS n-alcane torsion potential has been divided by 2 共see
Fig. 3兲. This was originally due to misprints in the original
articles,19,20 but we have included the results to show the
effect of the torsion potential for different molecules at different states.
The state points are distributed to cover the state space
extrema where transport properties may be compared with
experiment. For many state points and transport coefficients
we have chosen to compare it to the Assael correlation,12
which is known to represent all experimental data on viscosity, thermal conductivity, and diffusion to experimental accuracy as long as one does not extrapolate outside the region
of validity.
Before comparing the simulated data to experiment we
have used the temperature corrections ␣ described in Sec.
III. The corrections are all, except in five cases, smaller than
10%. The errors introduced by these corrections are therefore
less than 1% in addition to the estimated standard deviation
from the simulations themselves 共shown in Tables II–V兲. In
five cases the corrections are between 10% and 20%, thus
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7588
J. Chem. Phys., Vol. 112, No. 17, 1 May 2000
Dysthe, Fuchs, and Rousseau
TABLE V. 2-methylbutane transport coefficients.
␳
共kg m⫺3 )
␰
Experiment
SMMK
SMMK/2
610.0
0.558b
Experiment
SMMK
SMMK/2
750.0
Model
0.691b
␩
T
共K兲
p
共MPa兲
(10⫺3 Pa s兲
D
(10⫺9 m2 s⫺1 )
␭
共W m⫺1 K⫺1 )
320
323.3
320.1
6.2
8.875⫾0.194
6.164⫾0.178
0.185a
0.176⫾0.015
0.175⫾0.015
6.05⫾0.05
5.93⫾0.06
0.095⫾0.02
0.096⫾0.02
300
301.0
298.1
270
276.5⫾0.3
266.4⫾0.3
0.98a
0.78⫾0.07
0.65⫾0.06
1.521⫾0.032
1.565⫾0.016
0.17⫾0.03
0.18⫾0.03
From Collings and McLaughlin 共Ref. 34兲.
Packing fractions of n-pentane.
a
b
introducing an additional possible error of 1%–2%. Only in
one single case is the error introduced by the temperature
correction procedure comparable to the estimated standard
deviation from the simulations itself: for the diffusion coefficient of the AUA共2兲 model for n-butane at 732.3 kg m⫺3
and 150 K where both contribute 2% to the error estimate.
The deviations from experimental data are presented in
Figs. 4–7. The estimated errors are, in general, twice the size
of the symbols for ␩ and ␭ and smaller than the symbols for
D.
A. Simplicity and transferability
All the UA models are more or less equally simple to
program. A small complication is added in the computation
of the center of force position of the AUA models, but it
does not add significantly to the CPU time. The OPLS
model, which uses constrained bending angles, is, however,
generally much more time consuming due to the increased
number of iterations needed to converge with the RATTLE
algorithm. For n-decane we were forced to use 0.1 fs time
steps while compressing the system and only 2 fs during
production. For n-hexadecane the time steps had to be so
small (⬍1 fs兲 that the simulations became too CPUconsuming.
The AUA models are not constructed as transferable in
the same sense as the other UA models; the user has to
supply experimental bond lengths and bending angles for
any new alkane to be simulated. In practice, to simplify this
we have chosen to use the n-pentane parameters for shorter
alkanes and n-decane parameters for longer alkanes. Although this was not the original intention of Toxvaerd it puts
the AUA potentials on an equal footing with the other parameter sets for the sake of comparison.
FIG. 3. Torsion potentials used in this study. Solid line—AUA共2兲, long
dashed—AUA共3兲, dotted line—OPLS n-alkane, and dashed line—OPLS/2.
B. State and property independence
The UA models used in this work have been adjusted to
equilibrium properties. Comparing simulations of transport
properties with experiment is therefore a test of the property
independence of the models. As indicated in Sec. I and in
Fig. 1, the state points for the simulations in this work have
been carefully chosen to map out a maximum variation in
temperature and density within regions where experimental
data are available. In order to better grasp the main trends we
have plotted the deviations from experiment in Figs. 4–7 as
function of packing fraction, ␰ .
The packing fraction, ␰ , is the ratio of the density of the
fluid to the ‘‘close packed’’ density of the fluid. In the Enskog theory of hard spheres the transport coefficients diverge
as the density approaches the close packed density. When
applied to soft potentials the close packed density must be
temperature dependent. Assael et al.12 have used the Enskog
theory as a basis for correlating viscosity, thermal conductivity, and self-diffusion for n-alkanes and have found expressions for the close packed densities of n-alkanes such
that the reduced transport coefficients collapse to three universal curves. Because this correlation has utilized the universality of the Enskog theory and applied it to n-alkanes we
use their expressions to calculate the close packed densities.
One observes in Figs. 4–6 that for all molecules and all
UA models the viscosity computed deviates increasingly as
the packing fraction increases. The LJ results do not show
FIG. 4. Deviation from experiment of viscosity prediction as function of
packing fraction for n-decane for different states and models.
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J. Chem. Phys., Vol. 112, No. 17, 1 May 2000
Fluid transport properties by equilibrium molecular dynamics
7589
FIG. 5. Deviation from experiment of viscosity prediction as function of
packing fraction for n-butane for different states and models. Legends are as
in Fig. 4, in addition, the plus sign and dashed line: AA OPLS rigid and the
multiplication sign and dot dashed line: AA OPLS free rotation.
any clear trend, but unlike the UA results, the deviations stay
within 50%. The all atoms viscosities for n-butane of Allen
and Rowley11 show an increasing positive deviation as opposed to the negative deviation of the UA viscosities.
For the diffusion coefficient of n-decane one observes to
some degree the inverse trend as for the viscosity 共as expected兲: There is an increasing positive deviation as the
packing fraction increases. The results for the different models are more diverse than for the viscosity: The AUA and
SKS models deviate less than 20% from experiment whereas
SMMK and OPLS deviate up to 120%.
Both the magnitude and the spread of the deviations
共40%–80% for viscosity of n-decane, 55%–75% for viscosity of n-hexadecane, and 0%–200% for diffusion of
n-decane兲 at the highest packing fractions are surprisingly
large in light of the fact that both the ‘‘best’’ and the
‘‘worst’’ parameter sets have been carefully adjusted and
refined several times to reproduce equilibrium properties for
n-alkanes. Comparing the SKS and SMMK models one sees
that changing the ␴ CH3 by 4% and ␧ CH3 by 14% has had a
catastrophic effect on the prediction of transport properties.
Although the AUA model was proposed to take into account
the effect of anisotropy that should be more important at
high densities we cannot see any clear difference in the performance of the AUA共2兲 and SKS models even at high packing fractions.
FIG. 6. Deviation from experiment of viscosity prediction as a function of
packing fraction for 2-methylbutane and n-hexadecane for different states
and models. The legends are as in Fig. 4.
FIG. 7. Deviation from experiment of diffusion coefficient prediction as a
function of packing fraction for different fluids, states, and models. The
legends are as in Fig. 4.
One also observes that the two models with halved torsion potentials, SKS/2 and SMMK/2, deviate more from experiment than the models SKS and SMMK. The increased
deviation becomes more significant at high packing fractions
and for longer chain lengths, at the smallest packing fractions the deviations between SKS and SKS/2 and between
SMMK and SMMK/2 are not statistically significant. It thus
seems that torsionally stiff molecules are less mobile than
soft ones at high packing fractions. This is likely due to
either trans–gauche transitions or ‘‘wiggling’’ about a minimum, not to the slightly increased probability of being in the
gauche conformation. A ‘‘reptation’’-like picture of medium
length alkanes suggest that when the free volume decreases,
the barriers to conformation change becomes increasingly
important to the mobility. This is consistent with the findings
of Clarke and Brown.6 They compared n-hexane with and
without torsion potential at 200 and 300 K and found that
torsional fluctuations, not conformational transitions, were
coupled to the viscosity. The ability to ‘‘wiggle’’ around the
most favorable conformation at high density and low temperature increases the mobility of the individual molecules
and also the fluidity of the liquid as a whole. This points to a
possible route to improving the transport properties of the
UA models at high packing fractions without changing any
properties 共like the position of the critical point兲 at lower
packing fraction: make the torsion potential minima narrower while keeping the gauche energy and transition barriers in accordance with experimental data.
Comparing the different models one finds that the least
deviations are found for OPLS ( ␩ for n-butane兲 and AUA共2兲
( ␩ for n-decane and n-hexane, D for n-decane兲. The mean
deviations 共the mean over all states, diffusion, viscosity for
both n-butane and n-decane兲 are AUA共2兲 24%, SKS 26%,
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7590
J. Chem. Phys., Vol. 112, No. 17, 1 May 2000
AUA共3兲 30%, SKS/2 42%, OPLS 48%, SMMK 54% and
SMMK/2 99%. It should also be noted that the only model
which does not have an increasing deviation with the molecular chain length is AUA共2兲.
V. CONCLUSIONS
Viscosities are increasingly underestimated and diffusion coefficients are overestimated as the packing fraction is
increased. The same tendency is found with increasing chain
length except for the AUA共2兲 model. Clear evidence was
found that the torsion potential is very important for the
transport properties at high packing fractions. The existing
AUA models in their present form may be deemed slightly
less transferable than the other UA models but a new transferable AUA parameter set is in preparation.35 The interaction potential models have been adjusted to equilibrium
properties and have been shown to be acceptably accurate for
vapor liquid equilibria.36 The comparison of calculated transport coefficients with experiment is thus a test of the property independence of the models and we will rank the
‘‘goodness’’ of the interaction potential models based on the
criterion of state independence of transport coefficients:
AUA共2兲, SKS, AUA共3兲, OPLS, SMMK. The OPLS model,
however is more CPU intensive due to the number of constraints to be satisfied and is therefore considered as an impractical model for large molecules.
ACKNOWLEDGMENTS
We would like to thank Total Exploration Production for
a grant to DKD. We thank the Institut du Dévelopement et
des Ressources en Informatique Scientifique 共IDRIS兲 for a
generous allocation of Cray T3E computer time.
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