Vapor-Liquid Equilibria of Mixtures Containing

Alkanes, Carbon Dioxide, and Nitrogen

Jeffrey J. Potoff and J. Ilja Siepmann
Dept. of Chemistry and Dept. of Chemical Engineering and Materials. University of Minnesota, Minneapolis, M N 55455

New force fields for carboii dioxide and nitrogen are introduced that quantitatively
reproduce the vupor - liquid equilibiia (VLE) of the neat systems and their mixtures
with alkanes. [n addition to the usual VLE calculations for p~ireCO, and N,, calculu-
tions of the binary mixtures with propane were used in the force-field deeueiopment to
achieve a good balance between dispersive and electrostatic (quudrupole- quadrupole)
interactions. The transferability of the force fields was then assessed fiom calciilations
of the VLE for the binary mixtures with n-hexane, the binary mixture of CO,/N,,
- - and
the ternary mixture of C02/N2/propune. The VLE calculations were carried out using
configiirational-bias Monte Carlo simulations in either the grand canonical ensemble
with histogram- reweighting or in the Gibbs ensemble.

Introduction
The design of separation processes requires accurate to limit the predictive capability of EOS models to systems
knowledge of the vapor-liquid equilibria for each of the pure where experimental data are readily available.
components and the multicomponent mixture. Of particular Another promising alternative to EOS models for VLE
interest is the phase behavior of mixtures containing alkanes, calculations is molecular simulation. Over the last 10 years,
CO, and N,, which are typically found in the petrochemical the introduction of new simulation methodologies, such as
industry. One example is the flooding of petroleum reservoirs Gibbs ensemble Monte Carlo (GEMC) (Panagiotopoulos,
with CO, or N, to enhance oil recovery. Furthermore, signif- 1987; Panagiotopoulos et al., 1988), a recent molecular dy-
icant effort has been expended designing chemical processes namics variant of GEMC (Baranyai and Cummings. ZOOO),
where supercritical CO, is used as a “green”so1vent (Brunner, histogram-reweighting techniques (Ferrenberg and Swend-
1994; Mishima et al., 2000). This is due to the fact that CO, sen. 1988. 19891, and Gibbs-Duhem integration (Kofie, 1992).
has nearly ideal physical properties for such applications, in- have made the calculation of phase equilibria from simula-
cluding a low critical temperature of 304.3 K and moderate tion a routine task. If an accurate and transferable force field
critical pressure of 73.8 bar (Rainwater, 1991). is available, then molecular simulations have an advantage
Calculations of vapor-liquid equilibria (VLE) in chemical over EOS calculations in terms of their predictive capability,
engineering have traditionally been performed with an equa- which comes at the expense of a significantly higher expendi-
tion of state (EOS) (Reid et al., 1977). These EOS models ture of computer time necessary to determine a coexistence
are based on pure-phase properties, are easy to use, and re- curve.
quire little computational effort, but often yield only qualita- In this work, we demonstrate how molecular simulation
tive predictions of binary and ternary mixture VLE. For a techniques can be used to develop transferable force fields
quantitative representation of the VLE, however, it is usually and to predict the VLE of binary and ternary mixtures of
necessary to introduce empirically derived binary interaction substances relevant to common industrial processes. We fo-
parameters. This need for binary interaction parameters tends cus our attention on mixtures of nonpolar and (slightly) polar
molecules. The accurate prediction of VLE for these systems
has until now been possible only by fitting system-specific bi-
C‘orrcspondencr concerning this a r t i c l e s h o u l d he addressed to J . 1. Siupmonn. nary interaction parameters (Liu and Beck, 1998) or by using
Permanent a d d r e x of J. J . Potoft: Dept. of Chemical Engineermg and Materials
Science, Wayne State University. 5050 Anthony Wayne Drive. Drtroit. MI 48202. special combining rules for unlike-pair interactions on differ-

July 2001 Vol. 47, No. 7 AIChE Journal

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ent molecules (Potoff et al., 19991, that is the Lorentz- Table 1. Lennard-Jones Parameters and Partial Charges for
Berthelot combining rules (Maitland et al., 1981) were used the TraPPE Force Field
for unlike interactions of atoms belonging to the same type of
Site c/kh [Kl r [A1 q [el
molecule (such as methyl and methylene groups on propane),
C-H 15.3 3.31 0.0
whereas thc Kong combining rules (Kong, 1973; Maitland et C (in alkane CH, group) 4.0 3.30 0.0
al., 1Y81) were used for unlike-pair interactions of molecules C (in alkane CH2 group) 5.0 3.65 0.0
of different types (such as methyl group in propane and oxy- C (in CO,) 27.0 2.80 +0.70
gen atom in CO,). Use of different combining rules within 0 (in CO, ) 79.0 3.05 - 0.35

the same system is not entirely consistent and limits the N (in N,) 36.0 3.31 - 0.482

transferability of the force fields. The force field employed in

COM (in N?) 0.0 0.0 + 0.964
this work is the nonpolarizable all-atom transferable poten-
tials for phase equilibria (TraPPE) force field, which makes
use of Lennard-Jones potentials for the overlap and disper- imental bond length for long alkanes. Hydrogen interactio?
sive interactions, while simple point charges are used for the sites are placed at the midpoint of the C-H bonds (0.55 A
first-order electrostatic and induction interactions (Maitland away from the C site), which were found to yield a better
et al., 1981). The Lennard-Jones parameters for all unlike- representation of the alkanes than H sites at the H nuclei
pair interactions (irrespective of whether the sites belong to (Chen and Siepmann, 1999). Intramolecular flexibility is gov-
molecules of the same type or of different types) are deter- erned by bond angle bending and torsional potentials (Chen
mined from the commonly used Lorentz-Berthelot combining and Siepmann, 1999).
rules (Allen and Tildesley, 1987). Configurational-bias his-
togram-rewcighting Monte Carlo simulations in the grand
canonical ensemble were used to calculate pure-component Carbon dioxide
phase diagrams for CO, and N, and the phase diagrams for There have been a number of models proposed for CO,
the binary mixtures CO,/propane, CO,/n-hexane, CO,/N,, (Murthy et al., 1983; Harris and Yung, 1995; Vrabec and
and N,/propane. Gibbs ensemble Monte Carlo was used for Fischer, 1997; Potoff et al., 1999). However, only the EMP2
VLE calculations of the ternary mixture CO,/N,/propane. model by Harris and Yung (1995) and its exp-6 variant (Potoff
The remainder of this article is organized as follows. In the et al., 1999) were optimized for VLE calculations. While both
next section, the force fields used in this work are described the LJ and exp-6 forms of the EMP2 force field yield accu-
in detail. The computational details of this work are listed in rate pure-component VLE data, only a qualitative represen-
the third section. The results of our simulations are pre- tation of the binary mixture phase diagram for n-alkane/CO,
sented and discussed in the fourth section. Comparisons are mixtures is possible unless special combining rules are used
made between our newly developed models for CO, and N2 for the unlike-pair interactions (Potoff et al., 1999). To rectify
and others found in the literature. The conclusions of our this, we have developed our own CO, force field that is con-
work are given in the fifth section. sistent with the TraPPE-EH model for n-alkanes. This model
has three Lennard-Jones sites that model the overlap and
Models dispersion forces (Maitland et al., 1981). Partial point charges
are centered at each LJ site to approximate the first-order
Alkanes
electrostatic and second-order induction interactions (Mait-
The TraPPE-EH force field (Chen and Siepmann, 19991, land et al., 1981). The C - 0 bond length and O-CoO bond
which is based on an explicit-hydrogen representation, is used angle are fixed at the experimental value of 1.16 A and at
for propane and n-hexane. Nonbonded interaction sites in- 180", respectively. Tests of a model with a flexible bond angle
teract via a pairwise-additive Lennard-Jones 12-6 potential showed that the flexible and rigid models yield indistinguish-
able phase equilibrium properties. The parameter set for the
TraPPE CO, force field is listed in Table 1.

Nitrogen
where r , , . E , , , and a,, are the separation, LJ well depth, and As in the case of CO,, a number of intermolecular force
LJ size. respectively, for the pair of atoms i and j. The LJ fields for N, can be found in the literature (Murthy et al.,
parameters for interactions between like atoms are listed in 1983; Galassi and Tildesley, 1994). None of these force fields
Table 1. The parameters for unlike-pair interactions are cal- were optimized for VLE calculations, which is the motivation
culated with the Lorentz-Berthelot combining rules that are behind the development of the TraPPE N, force field. This
commonly used in molecular simulations (Allen and Tildes- force field consists of three sites. Each nitrogen atom is mod-
ley, 1987) eled by a Lennard-Jo2es site separated by the experimental
bond length of 1.10 A. The Lennard-Jones parameters for
these sites are listed in Table 1. The gas-phase quadrupole
moment of N, ( Q = - 1.4X esu) (Gray and Gubbins,
5, = 4% E,, ' (3)
1984) was reproduced by placing point charges of - 0.482 e
on each Lennard-Jones site. To maintain charge neutrality, a
point charge of +O.964 e was placed at the center of mass
C-C sites are separated by 1.54 A, a value close to the exper- (COM) of the N, molecule.

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Simulation details tions were truncated at 10 A with the appropriate long-range
Pure component and binary mixtures corrections applied, and the electrostatic interactions were
calculated via an Ewald summation technique with the same
Pure-component and binary-mixture phase diagrams were
parameters.
calculated from grand canonical histogram-reweighting
Monte Carlo simulations. The application of histogram-re-
weighting methods to binary mixture VLE calculations is de- Results
scribed in detail in the original references and is not re- Pure components
peated here (Potoff and Panagiotopoulos, 1998; Potoff et al.. We present two models for COz. The first is a minor varia-
1999). The Monte Carlo program employs rigid molecule tion of the EMP2 force field where parameters for the LJ
translations and rotations. conformational changes for the well depth and collision diameter were shifted slightly to
semiflexible alkanes, and molecule insertions and removals. compensate for a change in the C-0 bond length. In the
The coupled-decoupled configurational-bias Monte Carlo original model byoHarris and Yung (1995), an effective bond
method (Martin and Siepmann, 1999) was used for the latter length of 1.1490A was used, while the experimental bond
three types f: moves. Lennard-Jones interactions were trun- length of 1.16 A was used in this work to maintain consis-
cated at 10 A, and analytical long-range corrections were ap- tency between this model and the TraPPE models for the
plied to the energy, pressure, and chemical potential (Allen n-alkanes and nitrogen. The second model for CO, is an al-
and Tildesley, 1987). Electrostatic interactions were calcu- ternate parameterization derived by us (TraPPE CO, ). In
lated with the Ewald summation technique using tinfoil particular, the Lennard-Jones parameters and partial charges
boundary conditions, an Ewald screening parameter K X I, = were optimized to reproduce the vapor-liquid phase diagram
5, and K,,, = 5 for the upper bound of the reciprocal space of the binary mixture of CO,/propane. Thus, propane is es-
summation (Allen and Tildesley, 1987). sentially used as a “probe” molecule to find a suitable bal-
Pure-component simulations used a box length of L = 20 ance of the Lennard-Jones (overlap and dispersive) and
A (or 25 A near the critical region), while L = 25 A was used Coulombic (first-order electrostatic and induction) interac-
for the binary-mixture calculations. Critical points for the bi- tions. It should be noted, that this approach treats the inter-
nary mixtures were determined via a mixed-field analysis actions as effective interactions. that is, the Lennard-Jones
(Bruce and Wilding, 1992; Poioff and Panagiotopoulos, 1998) r p hpart implicitly includes many-body dispersive interactions
using a system size of L 25 A, except for the CO,/n-hexane and the additional contributions arising from instantaneous
mixture, where L = 30 A. The critical parameters reported dipole-quadrupole ( r - ‘ ) and quadrupole-quadrupole ( r - ‘“1
~
here are estimates for a single system size. To obtain more interactions. In a similar way, the partial charges mimic both
precise values of the critical parameters would require addi- the first-order electrostatic and second-order induction
tional and longer simulations for larger system sizes (Wilding, forces, and the TraPPE CO, quadrupole moment ( Q =
1995; Potoff and Panagiotopoulos, 1998). The final configura- - 4 . 5 2 ~lO-”esu) is slightly enhanced compared to the ex-
tion of a previous run was used as the initial configuration perimental value for an isolated CO, molecule ( Q = - 4.4 X
for another simulation at a different state point. Equilibra- 10p”esu) (Gray and Gubbins, 1984), but remains within the
tion periods of more than lo6 Monte Carlo steps (MCS) were statistical uncertainty of the experimental measurement.
used. Away from the critical region, vapor-phase simulations
were run for 5 x 10‘ MCS, while simulations in the liquid
phase were run for 1 to 2 X lo7 MCS. The production periods
were lengthened to 5 X lo7 MCS in the critical region. Simu-
lation data were stored in the form of lists containing the
number of molecules of each species and the energy of the
system. Samples were written to this list every 250 or 500
MCS.

Ternary mixtures
z
h

The GEMC technique (Panagiotopoulos, 1987; Pana- 260

giotopoulos et al., 1988) was used to determine the phase
behavior of the ternary mixture CO,/N,/propane. The more
direct GEMC technique avoids the need to construct a free-
240 11
energy surface over four-dimensional space, which would h m e
been required if the histogram-reweighting techniques had I
220
been used. While not impossible, histogram-reweighting tends 0 500 1000 1500
to become cumbersome to apply to systems of three or more kg/m3
components at conditions far from criticality.
The GEMC simulations were performed for N = 150 Figure 1. Vapor-liquid coexistence curve for carbon
molecules. Equilibration periods of l o 4 MC cycles (N at- dioxide.
Experimental coexistence d a t a a n d critical p i n t ( s o l ~ dlinc
tempted MC moves per cycle) were used and results were a n d astcrisk) (NIST, X O O ) , E M P Z model (squares). and
averaged over the next 5 X l o 4 MC cycles. As in the grand TraPPE model (circles); statistical unccrtaintics are smallcl
than t h e symbol sizes.
canonical Monte Carlo simulations, Lennard-Jones interac-
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4.5 I
3 -

m 2
-
I $2
~
s
3 -

1 -
2.5 -

3 32 3.4 3.6 3.8 4 4.2 4.4 0 ' I J

I
T 7 000 (I/K) 8 9 70 11 72
7 P 7 0 0 0 (7/K)
Figure 2. Clausius-Clapeyron plot of the saturated va-
por pressure for carbon dioxide vs. the in- Figure 4. Clausius-Clapeyron plot of the saturated va-
verse temperature. por pressure for nitrogen vs. the inverse tem-
f-xpcrimental data and critical point (solid line and asterisk) perature.
(NI5T 2000), EMP2 model (squares), and TraPPE model Experimental data and critical point (solid line and asterisk)
(ill'le\) (NIST, 2000). G T model (squares), and TraPPE model
icirclcs).

Much larger enhancements are a common feature for molec-

ular models that were fitted to describe the liquid state of 305.5 K, P, = 76.9 bar, pc = 462.5 kg/m3, respectively. Thus
dipolar fluids, for example, water (Jorgensen, 1983). both models yield excellent agreement with the experimental
The performance of these models with respect to the pre- data (Moldover and Gallagher, 1978; NIST, 2000): T, = 304.21
diction of vapor-liquid coexistence densities is compared in K, P, = 73.843 bar, and p, = 466.50 kg/m3.
Figure 1. Both models yield an equally accurate representa- In Figure 3, the vapor-liquid coexistence curves of N , for
tion of the phase diagram for neat CO,. The vapor pressure the TraPPE model and the GT model of Galassi and Tildes-
as a function of temperature is shown in Figure 2. Again, ley (1994) are shown. The TraPPE force field for N2 is in
both models predict the vapor pressure of CO, over the tem- good agreement with experimental data (NIST, 2000) with
perature rangz of interest equally well. The estimated critical respect to the saturated liquid and vapor densities over the
parameten for the TraPPE and modified EMP2 CO, models entire temperature range of interest. In contrast, the GT
are = 306.2 K. P, = 77.7 bar, p, = 464.9 kg/m3 and T, = model underpredicts the saturated vapor densities. As a re-
sult, the vapor pressure, shown in Figure 4, of the GT model
is approximately 15% too low over the entire temperature
1
130
n Q 1 range of interest. The TraPPE force field gives vapor pres-
sures that are in close agreement with experimental data from
85 K to 125 K. On the liquid side, the G T model overpredicts
the saturated liquid density at high temperatures, while un-
derpredicting the saturated liquid density at low tempera-
tures when compared to experiment. The critical parameters
are well reproduced by the TraPPE model [experimental val-
ues in brackets, taken from NIST (2000)l: T, = 126.5 K (126.19
K), P, = 34.6 bar (33.98 bar), pc = 308.0 kg/m3 (313.11
kg/m3).

80
0
F 200 400
kg/m3
600
I
800
Mixtures
As a further test of our models for CO, and N,, we per-
formed calculations for a number of binary mixtures. The first
of these mixtures was CO,/propane at T = 294.15 K, which
was also used in the parametrization of the TraPPE CO,
model (see earlier). In Figure 5, the results of the simulations
for the TraPPE force field are shown. For comparison, we
Figure 3. Vapor-liquid coexistence curve for nitrogen.
also include results for simulations of the EMP2 CO, model
Experimental coexistence data and critical point (solid line
and asterisk) ( N E T , ZOOO), G T model (squares), and TraPPE (Harris and Yung, 1995) with the TraPPE-EH propane model
model (circles). and the experimental data of Reamer et al. (1951). As an

AIChE Journal July 2001 Vol. 47, No. 7 1679

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80 1 ,

I
1 I

I ‘50
60
100

2 >
cu (ZI
2 40 0
Q

50
20

0
0 0.2 0.4 0.6 0.8 1 0 02 04 06 08 I
XC@*’ Yc**
Xm2p Ym*
Figure 6. Pressure-composition diagram for CO,/
Figure 5. Pressure-composition diagram for C O J
n-hexane at 353.15 K.
propane at 294.15 K.
txperiniental ddtli (ciicles) (1 I e t a l . 1981) m d FrdPPE
E x p e r i m e n t a l d a t a (circles) ( K e a r n e r e t al.. 1951), CO,/TraPPE-EH u-hesanc (solid line ,>lid d S t c r i \ k )
EMP?/TraPPE-EH propane (dashed line), TraPPE
COJTraPPE-EH propane (solid line).

pcrformed. Because thc pure-component results for the va-

aside, it should be noted here that the data of Reamer et al. por pressure of the G T force field showed significant devia-
are valid only below the critical temperature of CO, (Niesen tions from experiment, calculations were only performed for
a n d R a i n w a t e r , 1990). T h e c o m b i n a t i o n of t h e the TraPPE force field. In Figure 7, the pressure-composi-
EMP2/TraPPE-EH force fields is only in qualitative agree- tion diagram for CO,/N, at 253.15 K is shown. The simula-
ment with experiment, and exhibits significant deviations from tion results are in good agreement with the experimental
the experimental data for both the liquid and vapor phases. measurements (Gmehling and Onken, 1977) for the vapor
These deviations become increasingly severe as the concen- compositions, but the solubility of N2 in liquid CO, is under-
tration of CO, in the system increases. Simulation results for estimated slightly. The temperature used here is nearly twice
the TraPPE CO,/TraPPE-EH propane force fields, on the the critical temperature of NZ,and we speculate that the
other hand, are in good agreement with experiment over the functional form of the simple LJ potential can lead to inaccu-
entire composition range. Only slight deviations from experi- racies at these high reduced temperatures. Critical parame-
ment are found in the liquid phase. ters for this mixture were estimated from a mixed-field analy-
Another area of concern in the development of intermolec- sis to be P,. = 160.2 bar and xFoz = 0.612. Corresponding ex-
ular force fields is that of transferability. The parameters for perimental critical-point data are not availible.
the TraPPE CO, force field were originally tuned to improve The last of the binary mixtures studied was TraPPE
the CO,/propane binary-mixture phase behavior while re- N,/TraPPE-EH propane. In Figure 8, the pressure-composi-
taining the correct pure-component phase behavior of the tion diagram for N,/propane at 270 K is shown along with
EMP2 force field. If the model is “transferable,” then it the experimental results of Yucelen and Kidnay (1999). In
should yield phase behavior that is quantitatively in agree- the liquid phase, the simulation and experimental data are in
ment with experiment for any CO,/n-alkane mixture. A5 an good agreement. The simulation results show small devia-
example of this, simulations were performed for TraPPE tions from experiment in the vapor phase, as the solubility of
CO,/TraPPE-EH n-hexane. The pressure-composition dia- N, in liquid propane is underpredicted slightly. The esti-
gram for this system at 353.15 K is presented in Figure 6. mated critical parameters are P,. = 185.2 bar and xy2 = 0.626.
Over the entire composition range, the simulation data are in As in the case of the CO,/N, mixture, the corresponding
good agreement with the experimental data of Li et al. (1981). experimental critical parameters could not be found.
Only small deviations are present in the liquid phase, which We complete our study by presenting the phase diagram
is consistent with the observations made for the CO,/pro- for the ternary mixture of CO,/N,/propane at 270 K and 60
pane system. In addition, the estimated critical parameters of bar in Figure 9. The simulation data are in good agreement
P, = 109.9 bar and nFo’ = 0.86 agree rather well with their with experiment (Yuculen and Kidnay, 1999) for both the liq-
experimental counterparts of P, = 108.3 bar and xf302= 0.85 uid and vapor phases, demonstrating the predictive power of
(Li et al., 1981; Choi and Yeo, 1998). These results are a molecular simulation given an accurate and consistent set of
significant improvement over those of a recent Gibbs ensem- intermolecular forcc fields. The reader is reminded that in
ble study of CO,/n-hexane mixtures by Cui et al. (1999) where this work the Lorentz-Berthelot combining rules were used
for all unlike LJ interactions, and n o special binary interac-
united-atom alkane models were used.
Simulations for binary mixtures containing N? were also tion parameters were necessary to achieve quantitative agree-

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1680 July 2001 Vol. 47, No. 7
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200 7 1
’OX

0 100
100 75 50 25 0
I
0.2 0.4 0.6 0.8 1 Propane

”..,~ YM, Figure 9. Ternary phase diagram for COdNJpropane at

270 K and 60 bar.
Figure 7. Pressure-composition diagram for COdN, at Experimental data (squares) (Yucelen and Kidnay. 1999) a n d
253.15 K. TraPPE C O f l r a P P E N f l r a PPE- EH propane (diamonds).
Expcrimental data (circles) (Gmehling and Onken, 1977) and
TriiPPE C O f l r a P P E N, (solid line and asterisk).

for CO, and N,. These models were found to reproduce ex-
perimental results for the pure components to a high accu-
ment with experiment. This is a significant advance over other racy. For mixtures containing C 0 2 , our new model was found
recent studies of mixture-phase behavior via molecular simu- to yield results that are in quantitative agreement with exper-
lation, which have used special combining rules for unlike-pair iment, while previous simulations for CO,/alkane mixtures
interactions belonging to interaction sites on molecules of were not able to predict the correct phase behavior when
different types (Potoff et al., 1999) or have fitted system- Lorentz-Berthelot combining rules were consistently used.
specific binary interaction parameters (Vrabec and Fischer, Our CO, model was shown to be transferable, in the sense
1997; Liu and Beck, 1998). that the phase behavior of CO,/propane and CO,/n-hexane
are equally well reproduced. Simulations of mixtures of
Conclusions CO,,”,, however, showed small, but noticeable deviations
from experiment. As a result, further refinements might be
In this work we have developed the TraPPE force fields necessary for the N, force field, the most likely being the
addition of a third Lennard-Jones site at the bond center to
better tune the dispersive interactions and the repulsive shape
300 of the model. Finally, with the calculation of the ternary phase
diagram for CO,/N,/propane, we have shown that molecu-
lar simulation can be used to predict the phase behavior of
complex, multicomponent systems to a high accuracy without
requiring special binary interaction parameters.
200

> Acknowledgments

i/
m
s
a.
Financial support from the National Science Foundation, an Al-
fred P. Sloan Research Fellowship, and an MSI Research Scholar-
ship (J.J.P.) is gratefully acknowledged. Part of the computer re-
7 00 sources were provided by the Minnesota Supercomputing Institute
(MSI). We thank Bin Chen for stimulating discussions.

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