2578
J. Phys. Chem. B 1998, 102, 2578-2586
Thermodynamic Properties of the Williams, OPLS-AA, and MMFF94 All-Atom Force
Fields for Normal Alkanes
Bin Chen, Marcus G. Martin, and J. Ilja Siepmann*
Department of Chemistry, UniVersity of Minnesota, 207 Pleasant Street SE,
Minneapolis, Minnesota 55455-0431
ReceiVed: NoVember 6, 1997; In Final Form: January 23, 1998
The performance of several all-atom force fields for alkanes is compared and evaluated. Configurationalbias Monte Carlo simulations in the Gibbs ensemble were carried out to calculate the vapor-liquid phase
equilibria for methane, ethane, n-butane, n-pentane, and n-octane. The Williams, OPLS-AA, and MMFF94
force fields were selected as representative all-atom models for this study because they were fitted using
three different strategies (Williams, crystal structures and heats of sublimation; OPLS-AA, liquid densities
and heats of vaporization; MMFF94, rare gas pair potentials and quantum mechanics) and employ potentials
with three different functional forms to describe nonbonded van der Waals interactions (Williams, Buckingham
exp-r-6 ; OPLS-AA, Lennard-Jones 12-6; MMFF94, buffered 14-7). It is shown that seemingly small
differences in the potential functions can account for very large changes in the fluid-phase behavior. The
Williams and OPLS-AA force fields yield liquid densities, boiling temperatures, and critical points that are
in acceptable, albeit not in quantitative agreement with experiments, whereas the fluid-phase behavior of the
MMFF94 model shows very large deviations.
I. Introduction
Normal alkanes have been the focus of numerous computer
simulation studies for the past 20 years starting with the seminal
work of Ryckaert and Bellemans.1,2 The main reasons for the
great interest in alkanes are (i) alkanes are the prototypical case
of molecules with an articulated structure; (ii) alkanes are of
prime importance for the (petro-) chemical industries; and (iii)
alkyl groups are pivotal for the function of many biological
systems. The alkane models that have been used in molecular
simulations can be divided into two main categories: unitedatom (UA) models and all-atom (AA) models (sometimes also
referred to as explicit-hydrogen models). In the UA representation, methyl and methylene segments are treated as single
pseudoatoms with their interaction sites commonly located at
the position of the carbon atoms. Hydrogen atoms are treated
explicitly in AA models. The primary advantage of the UA
models is their computational efficiency; that is, by reducing
the number of interaction sites by a factor of 3, CPU time
savings of an order of magnitude can be achieved. On the other
hand, the AA models give a more faithful description of the
shape of real n-alkanes and are therefore considered to be more
appropriate for simulations of solid or high-density (lowtemperature) liquid phases.3,4 In addition, AA models allow
for the distribution of partial charges on the individual hydrogen
and carbon atoms, which may be important to describe the
interactions of alkanes with more polar molecules.
Jorgensen and co-workers5 have recently studied the performances of three AA force fields (OPLS-AA,6 AMBER94,7 and
MMFF948). They carried out Monte Carlo simulations in the
isobaric-isothermal ensemble for the liquid phase of n-butane
at its normal boiling point of -0.5 °C and reported that the
liquid density and heat of vaporization are well reproduced by
the OPLS-AA and AMBER94 force fields, whereas the MMFF94
* Corresponding author: [email protected].
force field did not yield a stable liquid phase at the selected
conditions. The aim of this paper is a more thorough investigation of the ability of three AA force fields to describe the fluidphase behavior of linear alkanes. The Williams,9 OPLS-AA,6
and MMFF948 force fields were selected as representative allatom models for this study because they were fitted using three
different strategies. The Williams model was parametrized
using the crystal structures and heats of sublimation for
n-pentane, n-hexane, and n-octane.9 For the OPLS-AA model
fits to the liquid densities and heats of vaporization for ethane,
propane, and n-butane were employed.6 Finally, Halgren8,10
used experimentally derived rare gas potentials and high-quality
ab initio calculations to obtain the parameters for his MMFF94
force field. In addition, the three force fields employ different
functional forms to describe the nonbonded van der Waals
interactions (Williams, Buckingham exp-r-6; OPLS-AA, Lennard-Jones 12-6; MMFF94, buffered 14-7) and use different
combining rules to determine the van der Waals interaction
parameters for the interactions of unlike atoms.
The remainder of this article is arranged as follows. Sections
2 and 3 give descriptions of the molecular force fields and
simulation details. In the first part of section 4, the influence
of the functional form used for nonbonded interactions on the
fluid-phase envelope is discussed. Thereafter, thermodynamic
and structural data for the three different force fields (and some
minor variations thereof) are presented. Finally, section 5
emphasizes some important points.
II. Force Fields
A. Intramolecular Interactions. The n-alkanes are treated
as semiflexible chain molecules in a way very similar to the
constrained molecular dynamics model used by Ryckaert and
co-workers.3 The lengths of all C-C and C-H bonds are fixed,
with parameters listed in Table 1. A harmonic bond bending
S1089-5647(98)00106-0 CCC: $15.00 © 1998 American Chemical Society
Published on Web 03/17/1998
All-Atom Force Fields for Alkanes
J. Phys. Chem. B, Vol. 102, No. 14, 1998 2579
TABLE 1: Bond Lengths, Equilibrium Bond Angles, and
Force Constants Used for the Williams, OPLS-AA, and
MMFF94 Force Fields
dC-C (Å)
dC-H (Å)
θ0(C-C-C) (deg)
kθ(C-C-C)/kB (K rad-2)
θ(C-C-H) (deg)
θ(H-C-H) (deg)
Williams
OPLS-AA
MMFF94
1.530
1.040
112.0
62556
112.5
106.0
1.529
1.090
112.7
58765
110.7
107.8
1.508
1.093
109.608
61637
110.549
108.836
potential is used for all C-C-C bond angles.11
ubend )
kθ
(θ - θ0)2
2
(1)
where kθ and θ0 are the bending force constant and the
equilibrium bond angle (the parameters are given in Table 1).
The H-C-H bond angles of all units and the C-C-H bond
angles containing a methyl group hydrogen are fixed, whereas
the C-C-H bond angles, for which the central carbon atom
belongs to a methylene segment, are determined via the same
geometric construction as used in ref 3. Thus, no contributions
to the intramolecular potential energy arise from H-C-H and
C-C-H bond angles. The OPLS united-atom torsional potential12 governs the motion of C-C-C-C dihedral angles,
utors(C-C-C-C) ) c1[1 + cos(φ)] + c2[1 - cos(2φ)] +
c3[1 + cos(3φ)] (2)
with the following numerical constants: c1/kB) 355.03 K, c2/
kB) -68.19 K, and c3/kB ) 791.32 K. The torsional motion
of dihedral angles involving the hydrogen atoms on a methyl
group is governed by a 3-fold symmetric torsional potential,
utors(X-C-C-H) ) cX[1 - cos(3φ)]
(3)
with cC/kB ) 854 K13 and cH/kB ) 717 K.14 There are no
potential energy contributions from torsional angles involving
methylene group hydrogens. Nonbonded van der Waals
interactions are included for all atoms that are separated by at
least three C-C bonds; that is, there are no 1-4 interactions
of any type. 1-5 interactions involving one hydrogen atom
and 1-6 interactions for two hydrogen atoms are also not
considered as nonbonded interactions, because they are accounted for by the torsional potentials. As mentioned above,
our choice of intramolecular potentials is closely related to the
constrained molecular dynamics model of Ryckaert and coworkers,3 who studied the rotator phase of n-triacosane using
the Williams force field. This choice is also computationally
very efficient for our Monte Carlo calculations (see section 3).
However, it differs from the more flexible models used
commonly in biomolecular simulations,15 which treat all bond
angles as flexible, explicitly include torsional potentials for all
dihedral angles, and often contain 1-4 nonbonded interactions.
Nevertheless, it is believed that the choice of intramolecular
potentials is of secondary importance for the determination of
thermodynamic properties, such as vapor-liquid phase equilibria, as long as it does not result in a change of the conformational characteristics of the molecule. We have carried out
some test simulations for more flexible representations, which
allowed all bond angles to vary, but found the influence of this
change on the fluid-phase behavior to be negligible (see section
5).
B. Intermolecular Potentials. As mentioned above, the
Williams, OPLS-AA, and MMFF94 force fields employ po-
Figure 1. Comparison of the nonbonded interaction potentials used
in the three force fields. The left, middle, and right set of curves are
for H-H, C-H, and C-C interactions, respectively. Solid, dashed,
and dotted lines represent Williams VII, OPLS-AA, and MMFF94
potentials, respectively.
tentials with different functional forms for the nonbonded van
der Waals interactions. The different van der Waals potentials
are compared graphically in Figure 1. The Williams force field9
makes use of Buckingham exp-6 potentials:16
uvdW(rij) ) Aij rij-6 + Bij exp(-Cijrij)
(4)
where rij is the separation between two atoms of types i and j.
The size parameters C are CCC ) 3.74 Å-1, CCH ) 3.67 Å-1,
and CHH ) 3.60 Å-1. Eight different combinations of A and B
parameters were proposed by Williams.9 Here, we used
parameter sets IV and VII, because set VII had been used earlier
for simulations of alkane crystals3 and set IV was recently found
to reproduce the experimental liquid densities of n-alkanes over
a wide range of chain lengths and temperatures.17 The
parameters for set IV are ACC/kB ) -2.858 × 105 K Å6, ACH/
kB ) -6.290 × 104 K Å6, AHH/kB ) -1.374 × 104 K Å6, BCC/
kB ) 4.208 × 107 K, BCH/kB ) 4.411 × 106 K, and BHH/kB )
1.335 × 106 K; those for set VII are ACC/kB ) -2.541 × 105
K Å6, ACH/kB ) -6.441 × 104 K Å6, AHH/kB ) -1.625 × 104
K Å6, BCC/kB ) 3.115 × 107 K, BCH/kB ) 5.536 × 106 K, and
BHH/kB ) 1.323 × 106 K.
The OPLS-AA force field6 utilizes Lennard-Jones 12-6
potentials18 for the van der Waals interactions:
[( ) ( ) ]
uvdW(rij) ) 4ij
σij
12
-
rij
σij
6
rij
(5)
with CC/kB ) 33.2 K, HH/kB ) 15.1 K, σCC ) 3.50 Å, and
σHH ) 2.50 Å. The parameters for unlike interactions are computed using standard Lorentz-Berthelot combining rules:19,20
1
σij ) (σii + σjj)
2
ij ) xiijj
(6)
In addition, a Coulombic interaction is used in the OPLS-AA
force field:
uq(rij) )
qiqj
4π0rij
(7)
where 0 is the permittivity of vacuum. The partial charge
2580 J. Phys. Chem. B, Vol. 102, No. 14, 1998
Chen et al.
carried by a hydrogen atom is qH ) +0.06e, while the charge
of a carbon atom depends on the number of hydrogens bonded
to it. Each methyl, methylene, or methane group is required to
carry no net charge; that is, qmethyl C ) -0.18e, qmethylene C )
-0.12e, and qmethane C ) -0.24e.
The MMFF94 force field is based on well-characterized
interaction potentials for rare gas atoms and ab initio calculations.8 Halgren10 proposed a buffered 14-7 potential for the
van der Waals interactions:
[
uvdW(rij) ) ij
][
1.07
(rij/r*
ij) + 0.07
7
]
1.12
-2
7
(rij/r*
ij) + 0.12
(8)
where r*ij and ij are the idealized minimum energy separation
and well depth for the interaction of atoms of types i and j.21
The interactions diameters for like atoms are calculated from
1/4
r*
ii ) AiRi
(9)
where R is the atomic polarizability and A is a numerical
constant. The diameters for unlike atoms are obtained from
[ (
[ ( ) ])]
2
r*
ii - r*
jj
1
r*ij ) (r*
+
r*
)
1
+
0.2
1
exp
-12
jj
2 ii
r*
ii + r*
jj
(10)
and the interaction strength is calculated from
ij )
-6
181.16GiGjRiRj(r*
ij)
(Ri /Ni)1/2 + (Rj /Nj)1/2
(11)
where G and N are a scale factor obtained for rare gas atoms
and the Slater-Kirkwood effective number of electrons, respectively. The resulting values for hydrogen and carbon atoms
are r*
CC ) 3.94 Å, r*
CH ) 3.60 Å, r*
HH ) 2.97 Å, CC/kB ) 34.12
K, CH/kB ) 14.13 K, and HH/kB ) 10.86 K. There are no
partial charges on carbons and hydrogens belonging to alkanes
in the MMFF94 force field.
III. Computational Details
A combination of the Gibbs ensemble Monte Carlo method
(GEMC)22-24 and the configurational-bias Monte Carlo (CBMC)
algorithm25-29 was employed to study the vapor-liquid phase
equilibria for five linear alkanes. The system sizes (number of
alkane molecules and combined volume of the two simulation
boxes) were chosen to yield liquid-phase simulation boxes with
linear dimensions of at least 20 Å and larger vapor phases
containing at least five molecules. Three hundred methane, 240
ethane, 150 n-butane, 120 n-pentane, or 90 n-octane molecules
were used in these simulations. For a given alkane, the initial
configuration was constructed as a layered crystal with all chains
in the all-trans conformation and oriented with their long axes
parallel to the z-axis of the Cartesian coordinate system. In
the x-y plane, the molecules were arranged in a hexagonal
structure with the backbone planes forming a herringbone
pattern.
Five different kinds of Monte Carlo moves were used to
sample phase space: translations of the center-of-mass, rotation
around the center-of-mass, conformational changes using CBMC,
volume exchanges between the two boxes, and CBMC particle
swaps between the two boxes. The maximum displacements
used for the translational, rotational, and volume moves were
adjusted to yield acceptance rates of 50%, where different
maximum translational and rotational displacements were used
for the vapor and liquid boxes.30 Several thousand Monte Carlo
cycles (N moves) at elevated temperature were used to “melt”
the initial structures, and the system was then “cooled” to the
desired temperature. During this initial phase of the equilibration procedure, only translational, rotational, and conformational
moves were employed. Thereafter, the systems were allowed
to equilibrate for at least 10 000 Monte Carlo cycles using all
five types of moves. The frequencies of swap and volume
moves were adjusted for this part and the production runs to
yield approximately one accepted swap and one accepted
volume move per 10 Monte Carlo cycles. The remainder of
moves were equally divided among the other three move types.
For each alkane, simulations were carried out for five to seven
different temperatures below the critical temperature. Most
simulations were started using the final configuration of a
neighboring temperature as the initial configuration (avoiding
the melting and cooling phases). Again, these simulations were
equilibrated for at least 10 000 Monte Carlo cycles. The results
reported in the next section were averaged over either 10 000
or 20 000 Monte Carlo cycles, and the statistical uncertainties
were estimated by dividing the simulations into 10 blocks.
A semiflexible representation of the alkanes was used for all
calculations, with the exception of one simulation for methane.
As mentioned earlier, all bond lengths are fixed throughout the
simulations. During the conformational and swap moves only
the carbon atoms are grown using the CBMC algorithm.26-29
The number of trial sites used during the CBMC growth is 4,
8, 8, and 12 for ethane, n-butane, n-pentane, and n-octane,
respectively. Ten trial sites are used for the insertion of the
first bead during a swap move.31,32 Once all carbon atoms are
successfuly generated, the positions of the methylene hydrogens
are obtained by a geometric procedure similar to that used to
enforce constraints in the molecular dynamics calculations.3
Since the C-H bond length and the methylene H-C-H angle
are fixed, forcing the H-C-H plane to be perpendicular to the
C-C-C plane and the H-C-H bisector to lie in the C-C-C
plane and point away from the methylene carbon is sufficient
to assign the methylene hydrogen positions. To calculate the
methyl hydrogen positions, a random C-C-C-H torsional
angle is selected from the correct distribution using a Boltzmann
rejection technique.33 The torsional angle then defines the
positions of the methyl group hydrogens using a C3 symmetry
and the fixed C-H bond length and methyl C-C-H angle.
For hydrogens on methane, the positions are determined by the
tetrahedral angle and C-H bond length with a randomly chosen
orientation. After all hydrogen positions are known, the
Lennard-Jones interactions of these are calculated and the
corresponding Boltzmann weight is used in addition to the
Rosenbluth weight for the acceptance of the CBMC move.
Similar procedures for the anisotropic Toxvaerd model or for
CBMC with two potential truncations have been described in
detail elsewhere.34,35
Spherical potential truncations at rcut ) 9 Å are used for all
nonbonded interactions. An atom-based truncation is used for
the van der Waals interactions, whereas a group-based truncation
is used for the Coulombic interactions.36 In this case, the
distance between the two carbon positions is used to decide
whether or not the interactions of an entire methane molecule,
methylene, or methyl group should be included. An additional
center-of-mass (COM)-based radial cutoff at rCOM ) rcut + dA+
dB, where dA is the maximum distances between any atom on
molecule A and its COM, is used to determine whether any
nonbonded interactions between molecules A and B have to be
computed.37 If not noted otherwise, long-range corrections were
All-Atom Force Fields for Alkanes
J. Phys. Chem. B, Vol. 102, No. 14, 1998 2581
Figure 2. Comparison of the Lennard-Jones 12-6 (solid line),
Buckingham exp-6 (dashed line), and buffered 14-7 (dotted line)
potential functions.
Figure 3. Comparison of the second virial coefficients obtained for
the Lennard-Jones 12-6 (solid line), Buckingham exp-6 (dashed line),
and buffered 14-7 (dotted line) potentials shown in Figure 2. T* is
the reduced temperature, i.e. T* ) T/(/kB).
applied to the van der Waals interactions. The usual analytical
tail corrections33 were used for the Lennard-Jones and Buckingham potentials. The energy and pressure tail corrections for
the buffered 14-7 potential were obtained from numerical
integration and are given by
3 U
ULC ) 2πNFXYr*
XY CXY
(12)
3 P
PLC ) -(2/3)πF2XYr*
XY C XY
(13)
where C UCC ) -0.024 84, C UCH ) -0.017 58, C UHH )
-0.008 377, C PCC ) 0.1696, C PCH ) 0.1204, and C PHH )
0.055 78 for rcut ) 9 Å (see also ref 38). A long-range
correction for the Coulombic interactions in the OPLS-AA force
field was not included in this work. As will be demonstrated
later, the small partial charges on the hydrogens and carbons
have only a negligible effect on thermodynamic properties, since
the architecture of the alkanes leads to an effective cancellation
of the charge-charge interactions between different molecules.
IV. Results and Discussion
A. Fluid-Phase Equilibria. Dependence on the Functional
Form of the Van der Waals Potential. To determine the
influence of the functional form of the van der Waals potential
on the fluid-phase behavior, we have first studied systems
consisting of single-bead particles. To allow for a meaningful
comparison, the parameters for the Buckingham exp-6 and
buffered 14-7 potentials were scaled to yield the same position
and depth of the potential energy minimum (and for the exp-6
potential also the same prefactor in the r-6 term) as the LennardJones potential. The three resulting potential functions are
shown in Figure 2. At first glance, the differences between the
three potential functions are very minor. The potential well of
the exp-6 potential is slightly wider than the well of the
Lennard-Jones 12-6 potential, where most of the difference is
in the repulsive region. In contrast, the buffered 14-7 potential
yields a narrower well compared to Lennard-Jones 12-6, and
most of the difference is in the attractive region. To what extent
can these apparently small differences affect thermodynamic
properties? As should be expected, the differences become more
pronounced when the second virial coefficients, B2, are compared (see Figure 3). The magnitude of B2 is approximately
Figure 4. Comparison of the vapor-liquid coexistence curves and
critical points obtained for the Lennard-Jones 12-6 (solid lines and
circles), scaled Buckingham exp-6 (dashed lines and diamonds), and
buffered 14-7 (dotted lines and triangles) potentials. The open symbols
are used for the coexistence densities obtained from the GEMC
simulations, the filled symbols are the estimates of the critical points,
and the lines are fits to the scaling law. T* and F* are the reduced
temperature and density.
20% smaller for the buffered 14-7 potential than for the exp-6
potential. The Lennard-Jones 12-6 potential falls in between,
but much closer to the exp-6 potential. Gibbs ensemble
simulations were carried out to determine the vapor-liquid
coexistence curves for single-bead systems interacting with the
three scaled potentials.39 The resulting phase diagrams are
shown in Figure 4. Again, the differences are appreciable. The
reduced critical temperatures are 1.401 ( 0.008, 1.297 ( 0.008,
and 1.126 ( 0.006 for the exp-6, Lennard-Jones 12-6, and
buffered 14-7 potential, respectively. Thus great care should
be taken when we want to compare force fields that are based
on different functional forms for the van der Waals interactions.
The potentials (such as plotted in Figure 1) will require different
well depths to yield the same thermodynamic properties. It
should also be noted that this will obviously lead to problems
when van der Waals potentials are fitted to ab initio energies.8,40
2582 J. Phys. Chem. B, Vol. 102, No. 14, 1998
TABLE 2: Details of the Variations of the Williams,
OPLS-AA, and MMFF94 Force Fields Used in This Work
W4
W7
W7-LC
OA
OA-q
MM
MM-LC
MMOPLS
vdW
parameters
partial
charges
vdW tail
corrections
Williams IV
Williams VII
Williams VII
OPLS-AA
OPLS-AA
MMFF94
MMFF94
MMFF94
no
no
no
yes
no
no
no
no
yes
yes
no
yes
yes
yes
no
yes
intramolecular
structure
Williams
Williams
Williams
OPLS-AA
OPLS-AA
MMFF94
MMFF94
OPLS-AA
Figure 5. Vapor-liquid coexistence curves for variations of the
Williams force field: (A) methane, ethane, and n-butane; (B) n-pentane
and n-octane. The solid lines and filled circles represent the experimental
coexistence densities and critical points.42,43 The calculated coexistence
densities and critical points are shown as diamonds, squares, and
triangles for the W4, W7, and W7-LC force fields, respectively.
Williams Force Field. Simulations were carried out for three
different variations of the Williams force field9 (see also Table
2): W4, Williams parameter set IV; W7, Williams parameter
set VII; W7-LC, same as W7 but without analytical tail
corrections for the vdW interactions. As mentioned earlier,
these two parameter sets were used in previous simulations for
n-alkanes.3,17 The vapor-liquid coexistence curves (VLCC) for
the Williams force field are shown in Figure 5, the corresponding numerical data are listed in Table S1, and the estimated
critical data and standard boiling temperatures are given in Table
3. The critical temperature and density are determined by leastsquares fits to the scaling law (using a scaling exponent of β )
0.32 ) and the law of rectilinear diameters24 (see ref 37 for a
detailed discussion of the uncertainties in the predicted critical
Chen et al.
properties that may arise from finite-size effects and from
changes in the scaling exponent), and the critical pressure and
boiling temperature are obtained from a Clausius-Clapeyron
plot.37,41 With the exception of methane, the VLCC for W4
and W7 are very similar and yield critical points that agree to
within the uncertainties of the calculations. For methane, W7
gives a critical temperature that is 5% higher than that for W4,
and the saturated liquid densities for W7 are shifted to higher
densities. As a result, the VLCC for W7 is in better agreement
with experiment.42,43 For ethane, the critical temperatures for
W4 and W7 are approximately 10% too low, which results in
a shift of the VLCC and too high vapor and too low liquid
densities. The saturated liquid densities of n-butane, n-pentane,
and n-octane for the W4 and W7 force fields are in good
agreement with the experimental results.43 However, while the
estimated critical temperatures for n-butane and n-pentane are
too low, that for n-octane is slightly too high. W7-LC performs
much less satisfactorily for all chain lengths; that is, tail
corrections should always be used with the Williams force field.
Here it should be noted that the energy parameters for the
Williams force field were fitted using relatively short potential
truncations, but assuming that the truncated potentials would
account for only 80% of the heat of sublimation.9
OPLS-AA Force Field. Simulations were carried out for two
versions of the OPLS-AA force field,6 which differed only with
respect to the use of partial charges (see also Table 2).
Jorgensen and co-workers6 reported that the partial charges have
a negligible effect on the liquid densities and heats of vaporization of pure alkanes. Thus we would like to investigate whether
the same holds true for the fluid-phase behavior. The VLCC
for the OPLS-AA force field are shown in Figure 6, the
corresponding numerical data are listed in Table S2, and the
estimated critical data and standard boiling temperatures are
given in Table 3.
As is evident from Figure 6 and Table 3, no significant
difference was found between simulations for the OPLS-AA
force field with (OA) and without (OA-q ) partial charges. For
both vapor and liquid phases Coulombic interactions contribute
less than 0.2% to the total nonbonded interactions; that is, the
structure of the normal alkanes leads to an effective cancellation
of the Coulombic interactions. A similar result has also been
observed for perfluorinated alkanes, where the partial charges
are much larger in magnitude.44
The OPLS-AA force field yields a very good prediction of
the critical and boiling temperatures of methane. However, the
saturated liquid densities and the critical density are overestimated by 10% and 5%, respectively. For all other n-alkanes,
the saturated liquid densities agree well with the experimental
data for the lower temperatures, which were used in the fitting
of the OPLS-AA force field.6 In contrast, the critical temperatures (and correspondingly also the liquid densities at higher
temperatures) are underestimated by as much as 6% for ethane
and 2% for n-octane. It might be of interest to note that our
calculations predict 270 K as the standard boiling temperature
for n-butane, which is 3 K lower than the experimental result.
Thus the stable phase of OPLS-AA n-butane at the experimental
boiling point is the vapor phase and not the liquid phase (see
ref 5).
Two additional tests were carried out for the OPLS-AA force
field. First, we have tested whether the limited flexibility used
in our simulations might be the cause for the disagreement in
the VLCC of methane. To this extent, simulations were carried
out for a methane model with flexible H-C-H bond angles
governed by a harmonic bending potential (see eq 1) with kθ/
All-Atom Force Fields for Alkanes
J. Phys. Chem. B, Vol. 102, No. 14, 1998 2583
TABLE 3: Thermodynamic Properties of n-alkanes for the Williams, OPLS-AA, and MMFF94 All-Atom Force Fields (Details
of the Force Fields As Described in Table 2) and the TraPPE United-Atom Force Field. Tb, Tc, Gc, and Pc Are the Normal
Boiling Point, the Critical Temperature, the Critical Density, and the Critical Pressure, respectively. The Experimental Data
Are Taken from Ref 42. The Subscripts Give the Statistical Accuracy of the Last Decimal(s)
W4
W7
W7-LC
Tb/K
Tc/K
Fc/(g cm)-3
Pc/MPa
1001
1722
0.1656
4.08
1022
1802
0.1625
4.04
951
1571
0.1726
3.96
Tb/K
Tc/K
Fc/(g cm)-3
Pc/MPa
1673
2744
0.2059
4.713
1692
2774
0.19510
3.812
Tb/K
Tc/K
Fc/(g cm)-3
Pc/MPa
2673
4105
0.23211
4.115
Tb/K
Tc/K
Fc/(g cm)-3
Pc/MPa
3012
4634
0.23311
3.57
Tb/K
Tc/K
Fc/(g cm)-3
Pc/MPa
OA-q
MM
MM-LC
TraPPE
expt
Methane
1141
1912
0.1706
4.69
1131
1912
0.1696
4.49
791
1232
0.1528
2.811
721
1142
0.1469
2.04
1121
1911
0.1603
4.511
112
191
0.162
4.6
1521
2413
0.2138
4.55
n-Ethane
1782
2852
0.2027
4.713
1732
2862
0.2057
4.211
1221
1821
0.2116
3.37
1151
1692
0.19410
2.910
1771
3042
0.2063
5.14
185
305
0.205
4.9
2662
4114
0.23510
3.99
2224
3455
0.23413
3.417
n-Butane
2702
4095
0.22411
3.89
2681
4074
0.22610
3.94
1852
2683
0.22313
2.410
1673
2383
0.24312
2.517
2612
4234
0.2316
4.14
273
425
0.228
3.8
3015
4614
0.23812
3.415
2614
3775
0.24216
3.523
n-Pentane
3014
4585
0.22514
3.313
3025
4564
0.22612
3.317
2103
2943
0.22713
2.110
1955
2623
0.23515
2.020
2962
4702
0.2384
3.71
309
470
0.230
3.4
n-Octane
3945
5588
0.23316
2.917
3992
5546
0.21714
2.54
3862
5702
0.2392
2.61
399
569
0.232
2.5
4023
57611
0.24521
2.911
OA
kB ) 33 234 K rad-2. In this case, hydrogens take part in
CBMC conformational changes and swap moves. The VLCC
for this model is also shown in Figure 6, and it is obvious that
this added H-C-H flexibility has little effect on the phase
behavior of methane.
Secondly, we have investigated the influence of the potential
truncation. Jorgensen and co-workers5,6,36 used a spherical
potential truncations at 11 Å, i.e., 2 Å larger than the value
used here. A set of calculations was carried out for n-pentane
using rcut ) 12 Å. The change in potential truncation was found
to cause no significant change in the VLCC (see Figure 6B)
and thermodynamic properties (see Table 4). Thus, the analytical tail corrections do adequately account for the interactions
in the range from 9 to 12 Å (see also discussion of the liquid
structures).
MMFF94 Force Field. Simulations were carried out for the
MMFF94 force field with and without tail corrections (see eqs
12 and 13). The calculated VLCC are shown in Figure 7, the
corresponding numerical data are listed in Table S3, and the
estimated critical data and standard boiling temperatures are
given in Table 3. As can be expected, the tail corrections for
the buffered 14-7 potential have a smaller, albeit still significant, effect than those for the longer ranged Lennard-Jones 12-6
and Buckingham exp-6 potentials (see Table 3).
The disagreement between VLCC obtained for the MMFF94
force field and the experimental results is striking. The predicted
critical temperature for ethane is lower than the experimental
result for methane, and the calculated VLCC for n-butane and
n-pentane fall below the experimental VLCC for ethane. For
example, T ) -0.5 °C, the experimental boiling point of
n-butane which was used by Kaminski and Jorgensen in their
comparison of all-atom force fields, is substantially above the
critical temperature of MM-LC butane.
What causes the dramatic failure of the MMFF94 force field
in describing the fluid-phase behavior of alkanes? Kaminski
and Jorgensen5 point to the different ratios of the C-C, C-H,
and H-H well depths between OPLS-AA and MMFF94.
However, the corresponding ratios differ by an even larger extent
for the Williams VII parameter set (see Figure 1), which
performs rather well. Thus there seems to be no magic ratio
for the interaction strengths (see also next section). OPLS-AA
and MMFF94 differ also to a large degree in C-C bond length
and C-C-C bond angle (see Table 1). To evaluate the
influence of the intramolecular structure on the phase behavior,
a set of calculations (MMOPLS) was carried out for n-pentane
using the MMFF94 parameters for the nonbonded interactions
and the OPLS-AA parameters for the intramolecular structure.
The longer bonds and wider angles of OPLS-AA lead to a larger
molar volume (lower density) for MMOPLS, but the boiling and
critical temperatures remain unchanged (see Figure 7B and Table
4).
Since neither the ratio between interaction strengths nor the
intramolecular structure seems to be responsible for the disagreement between MMFF94 and experiment, could the problem
be with the rare gas pair potentials that are the starting point in
the development of the MMFF94 force field? Figure 8 shows
a comparison of the experimental phase diagram for argon and
the calculated phase diagram using the buffered 14-7 potential
with the parameters listed in ref 10. In striking contrast to the
VLCC for the linear alkanes, the VLCC for argon is shifted to
higher temperatures and the buffered 14-7 potential yields an
overestimate of the critical temperature by approximately 8%.
However, this is exactly what we would like to see for a true
pair potential because adding the unfavorable three-body
Axilrod-Teller interactions45,46 should shift the calculated
coexistence curve into good agreement with the experimental
results. Thus judging from this test for argon, we feel that the
rare gas pair potentials are not responsible for the failure of the
MMFF94 force field. Here it should also be noted that the
original buffered 14-7 parameters for hydrogen, which were
2584 J. Phys. Chem. B, Vol. 102, No. 14, 1998
Chen et al.
Figure 6. Vapor-liquid coexistence curves for variations of the OPLSAA force field: (A) methane and ethane; (B) n-butane, n-pentane, and
n-octane. Experimental data are shown as in Figure 6. The calculated
coexistence densities and critical points are shown as diamonds and
squares for the OA and OA-q force fields, respectively. Triangles are
used to show the results for methane using the fully flexible OA model
and for n-pentane using rcut ) 12 Å.
Figure 7. Vapor-liquid coexistence curves for variations of the
MMFF94 force field: (A) methane and ethane; (B) n-butane and
n-pentane. Experimental data are shown as in Figure 6. The calculated
coexistence densities and critical points are shown as diamonds, squares,
and triangles for MM, MM-LC, and MMOPLS, respectively.
TABLE 4: Thermodynamic Properties of n-Pentane for the
OPLS-AA Force Field Using Two Different Potential
Truncations and for the MMFF94 Force Field Using Two
Different Intramolecular Structures. The Subscripts Give
the Statistical Accuracy of the Last Decimal(s)
Tb/K
Tc/K
Fc/(g cm)-3
Pc/MPa
OA (rcut ) 9 Å)
OA (rcut ) 12 Å)
MM
MMOPLS
3014
4585
0.22514
3.313
2964
4546
0.23014
3.414
2103
2943
0.22713
2.110
2141
2904
0.22414
2.36
derived by Halgren10 using an estimated atomic polarizability
and Slater-Kirkwood effective number of electrons, have a well
depth that is approximately 25% smaller in magnitude than the
final MMFF94 parameters, which were obtained using ab initio
data.8
Chain-Length Dependence of the Thermodynamic Properties.
Comparisons between the experimentally observed critical
temperatures, Tc, critical densities, Fc, and standard boiling
points, Tb, and the corresponding calculated quantities for the
three force fields are shown as functions of chain length in
Figures 9-11. The MMFF94 force field (with tail corrections)
underestimates Tc and Tb by approximately 30%. The Williams
VII force field underestimates Tc of methane and of ethane by
5% and 10%, respectively. For the medium-length alkanes the
agreement is better, but there is obviously a trend in Figures 9
and 11 that suggests that the Williams force field will yield too
Figure 8. Comparison of the vapor-liquid coexistence curve and
critical points for argon obtained for the buffered 14-7 potential (dotted
lines and triangles) and from experiment (solid lines and filled circle).47
The dotted lines are fits to the scaling law and the law of rectilinear
diameters.
high Tc and Tb for longer alkanes (nC > 8 ). Thus the
interactions of methylene segments are too strong in the
Williams force field. The OPLS-AA force field gives very good
agreement for Tc and Tb for methane, but underestimates Tc for
All-Atom Force Fields for Alkanes
Figure 9. Ratios of simulated versus experimental critical temperatures.
The following symbols are used for the different force fields: W7 (open
diamonds), W7-LC (filled diamonds), OA (open squares), OA-q (filled
squares), MM (open triangles), and MM-LC (filled triangles). For
comparison, the results for the TraPPE united-atom force field are
shown as crosses.
Figure 10. Ratios of simulated versus experimental critical densities.
Symbols as in Figure 8.
Figure 11. Ratios of simulated versus experimental normal boiling
temperatures. Symbols as in Figure 8.
ethane. The agreement between experiment and simulation
seems to improve with increasing length of the alkanes, which
suggests that the OPLS-AA force field accurately reproduces
the strength of the methylene interactions. The results for Fc
are harder to interpret, since there is more scatter and the relative
error on calculated Fc data is much larger than for the
J. Phys. Chem. B, Vol. 102, No. 14, 1998 2585
Figure 12. Liquid-phase site-site radial distribution functions for
n-pentane at 360 K (W4, W7, and OA) and at 220 K (MM). Solid,
dashed, dotted, and dashed-dotted lines are used for the W-4, W-7,
OA, and MM force fields, respectively. The radial distribution functions
for C-H and C-C pairs are vertically shifted by 1 and 2 units.
temperatures. For comparison, Tc, Fc, and Tb calculated for the
TraPPE force field,37 a simpler united-atom force field with
Lennard-Jones parameters fitted to VLCC for alkanes, are also
included in Table 3 and Figures 9-11. The TraPPE force field
yields better results for Tc (but this quantity was used in the
fitting procedure), whereas it performs with similar accuracy
as the OPLS-AA and Williams force fields for Fc and Tb.
However, because of the more accurate prediction of Tc, the
TraPPE force field yields better overall agreement with the
experimental VLCC.37 Thus at present, use of the computationally more demanding OPLS-AA and Williams all-atom force
fields does not yield gains in accuracy for the prediction of fluidphase equilibria of alkanes.
Here we should address two important observations. All three
force fields give larger deviations in Tc and Tb for ethane than
for methane or butane. This observation suggests that different
nonbonded vdW parameters are required for the carbon atom
in methane, in a methyl group, or in a methylene group for an
all-atom model to yield a quantitative description of thermodynamic properties; that is, the approximation that the vdW
parameters of an atom do not depend on the type of its bonded
neighbors6-10 is most likely not valid. The C-C well depth of
the Williams force field is much deeper than that of OPLSAA, while the H-H well is 2 times shallower (see Figure 1).
Considering the great differences between these two force fields,
it might be a surprise that they give relatively similar results.
From this we expect that there will be several combinations of
carbon and hydrogen vdW parameters that may work for allatom alkane models, particularly if we allow different carbon
parameters for different groups. Thus the empirical fitting of
vdW parameters for all-atom models will be a very difficult
task.
B. Liquid Structures. Radial distribution functions (RDFs)
of liquid-phase n-pentane are compared for the three force fields
in Figure 12. Since the VLCC for the MMFF94 force field is
very different, the RDFs are compared for temperatures that
yield similar liquid densities. At 360 K, the saturated liquid
densities for W4, W7, and OA are 0.556, 0.556, and 0.552 g
cm-3, and that for MM at 220 K is 0.567 g cm-3. Regarding
the liquid structure, the most obvious result is that there is very
little difference between the three force fields. RDFs were also
2586 J. Phys. Chem. B, Vol. 102, No. 14, 1998
calculated for other alkanes and other sets of temperatures, but
the differences between the RDFs were always very small. Thus
neither the different functional forms of the potential, the slightly
different positions of the well depths (see Figure 1), nor the
different bond lengths are sufficient to cause major differences
in liquid structure (if the comparison is made for similar liquid
densities, i.e. different temperatures for different models).
Therefore, we feel that RDFs or the corresponding radially
averaged structure factors are probably not a good means to
compare force fields for alkanes. Finally, at the temperatures
used in this work there is very little structure beyond the first
peak, and no peak is observed around 9 Å, the range of our
spherical potential truncation.
V. Conclusions
In this work, the performance of three all-atom force fields
for n-alkanes is assessed. While the Williams force field10 is
specific to n-alkanes, the OPLS-AA6 and MMFF948 force fields
are designed for more general purposes. Configurational-bias
Monte Carlo calculations in the Gibbs ensemble were used to
determine the vapor-liquid coexistence curves of methane,
ethane, n-butane, n-pentane, and n-octane. It is clearly evident
that the MMFF94 force field does not yield a good description
of the thermodynamic properties of the alkanes. The critical
temperatures are best reproduced by the OPLS-AA force field,
and in particular the agreement improves for longer alkanes,
whereas the Williams force field yields slightly better results
for the saturated liquid densities, in particular for methane. There
is no significant difference in fluid-phase behavior for the OPLSAA model with and without partial charges. All three force
fields show larger deviations from the experimental results for
ethane than for methane or butane, which suggests that specific
van der Waals parameters for carbon atoms belonging to methane, methyl, or methylene groups may be required. The large
differences between the Williams and OPLS-AA van der Waals
parameters combined with their similar fluid-phase behavior
illustrate that several combinations of parameters may work for
all-atom alkane models. Finally, the liquid structure is not very
useful to distinguish between the three alkane force fields.
Acknowledgment. We would like to acknowledge many
stimulating discussions with Doug Tobias, Mike Klein, JeanPaul Ryckaert, and Ian McDonald. Financial support from the
Petroleum Research Fund, administered by the American
Chemical Society (Grant No. 29960-AC9), through a Camille
and Henry Dreyfus New Faculty Award, and through a
McKnight/Land-Grant Fellowship is gratefully acknowledged.
M.G.M. would like to thank the Graduate School, University
of Minnesota, for the award of a Graduate School Fellowship.
Part of the computer resources were provided by the Minnesota
Supercomputer Institute through the University of MinnesotaIBM Shared Research Project and NSF Grant CDA-9502979.
Supporting Information Available: Tables S1, S2, and S3
list the numerical results (pressure, vapor and liquid densities)
for the Williams, OPLS-AA, and MMFF94 force fields,
respectively (6 pages). Ordering information is given on any
current masthead page.
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