The prediction of the equation of state and the phase behavior of simple fluids (noble gases, carbon dioxide, benzene, methane, and short alkane chains) and their mixtures by Monte Carlo computer simulation and analytic approximations based on thermodynamic perturbation theory is discussed. Molecules are described by coarse grained models, where either the whole molecule (carbon dioxide, benzene, and methane) or a group of a few successive CH2 groups (in the case of alkanes) are lumped into an effective point particle. Interactions among these point particles are fitted by Lennard–Jones (LJ) potentials such that the vapor-liquid critical point of the fluid is reproduced in agreement with experiment; in the case of quadrupolar molecules a quadrupole-quadrupole interaction is included. These models are shown to provide a satisfactory description of the liquid-vapor phase diagram of these pure fluids. Investigations of mixtures, using the Lorentz–Berthelot (LB) combining rule, also produce satisfactory results if compared with experiment, while in some previous attempts (in which polar solvents were modeled without explicitly taking into account quadrupolar interaction), strong violations of the LB rules were required. For this reason, the present investigation is a step towards predictive modeling of polar mixtures at low computational cost. In many cases Monte Carlo simulations of such models (employing the grand-canonical ensemble together with reweighting techniques, successive umbrella sampling, and finite size scaling) yield accurate results in very good agreement with experimental data. Simulation results are quantitatively compared to an analytical approximation for the equation of state of the same model, which is computationally much more efficient, and some systematic discrepancies are discussed. These very simple coarse-grained models of small molecules developed here should be useful, e.g., for simulations of polymer solutions with such molecules as solvent.

1.
D.
Levesque
,
J.-J.
Weis
, and
J. P.
Hansen
, in
Monte Carlo Methods in Statistical Physics
, edited by
K.
Binder
(
Springer
,
Berlin
,
1979
), p.
47
.
2.
D.
Levesque
,
J.-J.
Weis
, and
J. P.
Hansen
, in
Applications of the Monte Carlo Method in Statistical Physics
, edited by
K.
Binder
(
Springer
,
Berlin
,
1984
), p.
37
.
3.
D.
Levesque
and
J.-J.
Weis
, in
The Monte Carlo Method in Condensed Matter Physics
, edited by
K.
Binder
(
Springer
,
Berlin
,
1992
), p.
121
.
4.
D.
Frenkel
and
B.
Smit
,
Understanding Molecular Simulation: From Algorithms to Applications
(
Academic
,
San Diego
,
1996
).
5.
N. B.
Wilding
, in
Computer Simulations in Condensed Matter: From Materials to Chemcial Biology
, edited by
M.
Ferrario
,
G.
Ciccotti
, and
K.
Binder
(
Springer
,
Berlin
,
2006
), Vol.
1
, p.
39
;
M.
Müller
and
J. J.
de Pablo
, ibid., p.
67
.
6.
A. Z.
Panagiotopoulos
,
Mol. Phys.
61
,
813
(
1987
).
7.
A. Z.
Panagiotopoulos
,
N.
Quirke
,
M.
Stapleton
, and
D. J.
Tildesley
,
Mol. Phys.
63
,
527
(
1988
).
8.
A. Z.
Panagiotopoulos
,
Mol. Simul.
9
,
1
(
1992
).
9.
J. L.
Siepmann
and
D.
Frenkel
,
Mol. Phys.
75
,
59
(
1992
).
10.
D.
Frenkel
,
G. C. A. M.
Mooji
, and
B.
Smit
,
J. Phys.: Condens. Matter
4
,
3053
(
1992
).
11.
J. J.
de Pablo
,
M.
Laso
, and
U. W.
Suter
,
J. Chem. Phys.
96
,
2395
(
1992
).
12.
A. M.
Ferrenberg
and
R. H.
Swendsen
,
Phys. Rev. Lett.
61
,
2635
(
1988
).
13.
A. M.
Ferrenberg
and
R. H.
Swendsen
,
Phys. Rev. Lett.
63
,
1195
(
1989
).
14.
R. H.
Swendsen
,
J. S.
Wang
, and
A. M.
Ferrenberg
, in
The Monte Carlo Method in Condensed Matter Physics
, edited by
K.
Binder
(
Springer
,
Berlin
,
1992
), p.
75
.
15.
M. E.
Fisher
, in
Critical Phenomean
, edited by
M. S.
Green
(
Academic
,
London
,
1971
), p.
1
.
16.
K.
Binder
,
Z. Phys. B
43
,
119
(
1981
).
17.
K.
Binder
, in
Computational Methods in Field Theory
, edited by
C. B.
Lang
and
H.
Gausterer
(
Springer
,
Berlin
,
1992
).
18.
K.
Binder
,
Rep. Prog. Phys.
60
,
487
(
1997
).
19.
N. B.
Wilding
and
A. D.
Bruce
,
J. Phys.: Condens. Matter
4
,
3087
(
1992
).
20.
N. B.
Wilding
,
Phys. Rev. E
52
,
602
(
1995
).
21.
N. B.
Wilding
,
J. Phys.: Condens. Matter
9
,
585
(
1997
).
22.
J.
Perez-Pellitero
,
P.
Ungerer
,
G.
Orkoulas
, and
A. D.
Mackie
,
J. Chem. Phys.
125
,
054515
(
2006
).
23.
G. M.
Torrie
and
J. P.
Valleau
,
J. Comput. Phys.
23
,
187
(
1977
).
24.
P.
Virnau
and
M.
Müller
,
J. Chem. Phys.
120
,
10925
(
2004
).
25.
B. A.
Berg
and
T.
Neuhaus
,
Phys. Lett. B
267
,
249
(
1991
);
B. A.
Berg
and
T.
Neuhaus
,
Phys. Rev. Lett.
68
,
9
(
1992
).
[PubMed]
26.
A. P.
Lyubartsev
,
A. A.
Martsinovska
,
S. V.
Shorbunov
, and
P. N.
Vorontsov-Velyaminov
,
J. Chem. Phys.
96
,
1776
(
1992
).
27.
F. A.
Escobedo
and
J. J.
de Pablo
,
J. Chem. Phys.
105
,
4391
(
1996
).
28.
H. J.
Böhm
and
R.
Ahlrichs
,
Mol. Phys.
55
,
445
(
1985
).
29.
M. G.
Martin
and
J. I.
Siepmann
,
J. Phys. Chem. B
103
,
4508
(
1999
);
M. G.
Martin
and
J. I.
Siepmann
,
J. Phys. Chem. B
102
,
2569
(
1998
).
30.
R.
Bukowski
,
J.
Sadlej
,
B.
Jeziorski
,
P.
Jankowski
,
K.
Szalewicz
,
S. A.
Kucharski
,
H. L.
Williams
, and
B. M.
Rice
,
J. Chem. Phys.
110
,
3785
(
1999
).
31.
S.
Bock
,
E.
Bich
, and
E.
Vogel
,
Chem. Phys.
257
,
147
(
2000
).
32.
Z.
Zhang
and
Z.
Duan
,
J. Chem. Phys.
122
,
214507
(
2005
).
33.
Supercritical Fluids
, edited by
E.
Kiran
and
J. M. H.
Levelt-Sengers
(
Kluwer
,
Dordrecht
,
1994
).
34.
Supercritical Carbon Dioxide in Polymer Reaction Engineering
, edited by
M. F.
Kemmere
and
Th.
Meyer
(
Wiley-VCH
,
Weinheim
,
2005
).
35.
B. M.
Mognetti
,
L.
Yelash
,
P.
Virnau
,
W.
Paul
,
K.
Binder
,
M.
Müller
, and
L. G.
MacDowell
,
J. Chem. Phys.
128
,
104501
(
2008
).
36.
B. M.
Mognetti
,
M.
Oettel
,
L.
Yelash
,
P.
Virnau
,
W.
Paul
, and
K.
Binder
,
Phys. Rev. E
77
,
041506
(
2008
).
37.
R. L.
Scott
and
P. H.
van Konynenburg
,
Discuss. Faraday Soc.
49
,
87
(
1970
).
38.
P. H.
van Konynenburg
and
R. L.
Scott
,
Philos. Trans. R. Soc. London, Ser. A
298
,
495
(
1980
).
39.
J. S.
Rowlinson
and
F. L.
Swinton
,
Liquids and Liquid Mixtures
(
Butterworths
,
London
,
1982
).
40.
A.
Bolz
,
U. K.
Deiters
,
C. J.
Peters
, and
T. W.
de Loos
,
Pure Appl. Chem.
70
,
2233
(
1998
).
41.
P.
Virnau
,
M.
Müller
,
L. G.
MacDowell
, and
K.
Binder
,
J. Chem. Phys.
121
,
2169
(
2004
).
42.

Indeed using results of Ref. 35 the CG parameters for given substances (with a reasonable quadrupolar moment) can be computed in a straightforward way without additional simulations.

43.
Coarse-Graining of Condensed Phase and Biomolecular Systems
, edited by
G.
Voth
(
Taylor and Francis
,
London
, 2009).
44.
Handbook of Material Modeling
, edited by
S.
Yip
(
Springer
,
Berlin
,
2005
).
45.
Simulation Methods for Polymers
, edited by
M. J.
Kotelyanskii
and
D. Y.
Theodorou
(
Dekker
,
New York
,
2004
).
46.

However, sometimes it is far from being clear if deviations from LB combining rules are in fact compensation of some bias of the model used. For instance in the case of polar substances the present work shows very clearly how strong violations of the rules in Ref. 41 were more properly due to a bad modeling for the solvents.

47.
C. P.
Hicks
,
R. L.
Hurle
, and
C. L.
Young
,
J. Chem. Soc., Faraday Trans. 2
73
,
1884
(
1977
).
48.
E. A.
Müller
and
L. D.
Gelb
,
Ind. Eng. Chem. Res.
42
,
4123
(
2003
).
49.
J.
Stoll
,
J.
Vrabec
, and
H.
Hasse
,
AIChE J.
49
,
2187
(
2003
).
50.
J.
Vrabec
,
J.
Stoll
, and
H.
Hasse
,
Mol. Simul.
31
,
215
(
2005
).
51.
J.
Vrabec
,
J.
Stoll
, and
H.
Hasse
,
J. Phys. Chem. B
105
,
12126
(
2001
).
52.

It is interesting to observe how atomistic models that are not improved to describe all the pure substances phase diagram but take as experimental input only the critical temperature and density (like in our case) are less accurate than our simpler model (see for instance the discussion of EPM2 model (Ref. 53) in Ref. 35).

53.
J. G.
Harris
and
K. H.
Yung
,
J. Phys. Chem.
99
,
12021
(
1995
).
54.
J. J.
Potoff
,
J. R.
Errington
, and
A. Z.
Panagiotopoulos
,
Mol. Phys.
97
,
1073
(
1999
).
55.
J.
Delhommelle
,
A.
Boutin
, and
A. H.
Fuchs
,
Mol. Simul.
22
,
351
(
1999
).
56.
J. J.
Potoff
and
J. I.
Siepmann
,
AIChE J.
47
,
1676
(
2001
).
57.
P.
Virnau
,
M.
Müller
,
L. G.
MacDowell
, and
K.
Binder
,
Comput. Phys. Commun.
147
,
378
(
2002
).
58.
P.
Virnau
,
M.
Müller
,
L. G.
MacDowell
, and
K.
Binder
,
New J. Phys.
6
,
7
(
2004
).
59.
K.
Binder
,
Phys. Rev. A
25
,
1699
(
1982
).
60.
J. E.
Hunter
 III
and
W. P.
Reinhardt
,
J. Chem. Phys.
103
,
8627
(
1995
).
61.
J.
Potoff
and
A.
Panagiotopoulos
,
J. Chem. Phys.
112
,
6411
(
2000
).
62.
R. L. C.
Vink
and
J.
Horbach
,
J. Phys.: Condens. Matter
16
,
3807
(
2004
).
63.
L.
Yelash
,
P.
Virnau
,
W.
Paul
,
K.
Binder
, and
M.
Müller
Phys. Rev. E
78
,
031801
(
2008
).
64.
G. C.
Maitland
,
M.
Rigby
,
E. B.
Smith
, and
W. A.
Wakeham
,
Intramolecular Forces, Their Origin and Determination
(
Clarendon
,
Oxford
,
1981
).
65.
K.
Binder
,
M.
Müller
,
P.
Virnau
, and
L. G.
MacDowell
,
Adv. Polym. Sci.
173
,
1
(
2005
).
66.
M. S.
Wertheim
,
J. Chem. Phys.
87
,
7323
(
1987
).
67.
L. G.
MacDowell
,
M.
Müller
,
C.
Vega
, and
K.
Binder
,
J. Chem. Phys.
113
,
419
(
2000
).
68.
L. G.
MacDowell
,
P.
Virnau
,
M.
Müller
, and
K.
Binder
,
J. Chem. Phys.
117
,
6360
(
2002
).
69.
J. P.
Hansen
and
I. R.
McDonald
,
Theory of Simple Liquids
(
Academic
,
New York
,
1986
).
70.
J. A.
Barker
and
D.
Henderson
,
J. Chem. Phys.
47
,
4714
(
1967
).
71.
Y.
Tang
and
B. C.-Y.
Lu
,
J. Chem. Phys.
99
,
9828
(
1993
).
72.
Y.
Tang
,
Z.
Tong
, and
B. C.-Y.
Lu
,
Fluid Phase Equilib.
134
,
21
(
1997
).
73.
A.
Parola
,
D.
Pini
, and
L.
Reatto
,
Phys. Rev. Lett.
100
,
165704
(
2008
).
74.
A.
Parola
,
D.
Pini
, and
L.
Reatto
,
Phys. Rev. Lett.
53
,
2417
(
1984
).
75.
W. G.
Chapman
,
K. E.
Gubbins
,
G.
Jackson
, and
M.
Radosz
,
Fluid Phase Equilib.
52
,
31
(
1989
);
L.
Yelash
,
M.
Müller
,
W.
Paul
, and
K.
Binder
,
Phys. Chem. Chem. Phys.
7
,
3728
(
2005
).
[PubMed]
76.
L. D.
Landau
and
E. M.
Lifshitz
,
Statistical Physics
, 3rd. ed.(
Butterworth-Heinemann
,
Oxford
,
1996
).
78.
M. W.
Pestak
,
R. E.
Goldstein
,
M. H. W.
Chan
,
J. R.
de Bruyn
, and
N. W.
Ashcroft
,
Phys. Rev. B
36
,
599
(
1987
).
79.
G.
Raabe
and
R. J.
Sadus
,
J. Chem. Phys.
119
,
6691
(
2003
).
80.
L.
Wang
and
R. J.
Sadus
,
J. Chem. Phys.
125
,
144509
(
2006
).
81.
J. C. G.
Calado
,
E.
Chang
, and
W. B.
Streett
,
Physica A
117
,
127
(
1983
).
82.
P. S.
Arora
,
H. L.
Robjohns
, and
P. J.
Dunlop
,
Physica A
95
,
561
(
1979
).
83.
M.
Klein
and
H. J. M.
Hanley
,
J. Chem. Phys.
53
,
4722
(
1970
).
84.
H. J. M.
Hanley
and
M.
Klein
,
J. Phys. Chem.
76
,
1743
(
1972
).
85.
M.
Saharay
and
S.
Balasubramanian
,
J. Chem. Phys.
120
,
9694
(
2004
).
86.
Y.
Zhang
,
J.
Yang
, and
Y.-X.
Yu
,
J. Phys. Chem. B
109
,
13375
(
2005
).
87.
G.
Stell
,
J. C.
Rasiaiah
, and
H.
Narang
,
Mol. Phys.
27
,
1392
(
1974
).
88.
K.
Binder
, in
Monte Carlo and Molecular Dynamics Simulations in Polymer Science
, edited by
K.
Binder
(
Oxford University Press
,
New York
,
1995
), p.
1
.
89.
B.
Smit
,
J. L.
Siepmann
, and
S.
Karaborni
,
J. Chem. Phys.
102
,
2126
(
1995
).
90.
K.
Kremer
and
G. S.
Grest
,
J. Chem. Phys.
92
,
5057
(
1990
).
91.
K. S.
Pitzer
,
J. Am. Chem. Soc.
77
,
3427
(
1955
).
92.
L.
Zhang
and
J. I.
Siepmann
,
J. Phys. Chem. B
109
,
2911
(
2005
).
93.
C. L.
Kong
,
J. Chem. Phys.
59
,
2464
(
1973
).
94.
B. E.
Fender
and
G. D.
Halsey
, Jr.
,
J. Chem. Phys.
36
,
1881
(
1962
).
95.
F. T.
Smith
,
Phys. Rev. A
5
,
1708
(
1972
).
96.
C.
Menduiña
,
C.
McBride
, and
C.
Vega
,
Phys. Chem. Chem. Phys.
3
,
1289
(
2001
).
97.
J.
Wang
,
C. A.
Cerdeirina
,
M. A.
Anisimov
, and
J. V.
Sengers
,
Phys. Rev. E
77
,
031127
(
2008
).
98.
M. E.
Fisher
and
G.
Orkoulas
,
Phys. Rev. Lett.
85
,
696
(
2000
).
99.
G.
Orkoulas
,
M. E.
Fisher
, and
C.
Üstün
,
J. Chem. Phys.
113
,
7530
(
2000
).
100.
Y. C.
Kim
,
M. E.
Fisher
, and
G.
Orkoulas
,
Phys. Rev. E
67
,
061506
(
2003
).
101.
B. H.
Sage
,
W. N.
Lacey
, and
J. G.
Schaafsma
,
Ind. Eng. Chem.
26
,
214
(
1934
).
102.
B. H.
Sage
,
R. H.
Olds
, and
W. N.
Lacey
,
Ind. Eng. Chem.
34
,
1108
(
1942
).
103.
T.-C.
Chu
,
R. J. J.
Chen
,
P. S.
Chappelaer
, and
R.
Kobayashi
,
J. Chem. Eng. Data
21
,
41
(
1979
).
104.
H.-M.
Lin
,
H. M.
Sebastian
,
J. J.
Simnick
, and
K.-C.
Chao
,
J. Chem. Eng. Data
24
,
146
(
1979
).
105.
H.
Cheng
,
M. E. P.
de Fernandez
,
J. A.
Zollweg
, and
W. B.
Streett
,
J. Chem. Eng. Data
34
,
319
(
1989
).
106.
G.
Schneider
,
Z.
Alwani
,
W.
Heim
,
E.
Horvath
, and
E. U.
Franck
,
Chem.-Ing.-Tech.
39
,
649
(
1967
).
107.
J. D.
Hottovy
,
K. D.
Luks
, and
J. P.
Kohn
,
J. Chem. Eng. Data
26
,
256
(
1981
).
108.
H. M.
Sebastian
,
J. J.
Simnick
,
H.-M.
Lin
, and
K.-C.
Chao
,
J. Chem. Eng. Data
25
,
138
(
1980
).
109.
B.
Garzon
,
S.
Lago
,
C.
Vega
,
E.
de Miguel
, and
L. F.
Rull
,
J. Chem. Phys.
101
,
4166
(
1994
).
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