Statistical associating fluid theory for chain molecules with attractive
potentials of variable range
Alejandro Gil-Villegas, Amparo Galindo, Paul J. Whitehead, Stuart J. Mills, and George
Jackson
Department of Chemistry, University of Sheffield, Sheffield, S3 7HF, United Kingdom
Andrew N. Burgess
Research and Technology, ICI Chemicals and Polymers, PO Box 8, The Heath, Runcorn, Cheshire,
WA7 4QD, United Kingdom
~Received 28 August 1996; accepted 30 October 1996!
A version of the statistical associating fluid theory ~SAFT! is developed for chain molecules of
hard-core segments with attractive potentials of variable range ~SAFT-VR!. The different
contributions to the Helmholtz free energy are evaluated according to the Wertheim perturbation
theory. The monomer properties are obtained from a high-temperature expansion up to second
order, using a compact expression for the first-order perturbation term ~mean-attractive energy!
a 1 . Making use of the mean-value theorem, a 1 is given as the van der Waals attractive constant and
the Carnahan and Starling contact value for the hard-sphere radial distribution function in terms of
an effective packing fraction. The second-order perturbation term a 2 is evaluated with the local
compressibility approximation. The monomer cavity function, required for the calculation of the
free energy due to the formation of the chains and the contribution due to association, is given as
a function of a 1 . We analyse the equation of state for chain molecules with three different types of
monomer hard-core potentials with variable attractive range: square-well ~SW!, Yukawa ~Y!, and
Sutherland ~S!. The theory for the hard-core potentials can easily be generalised to soft-core
systems: we develop a simple equation of state for Mie m2n potentials, of which the Lennard-Jones
~LJ! 6-12 potential is a particular case. The equations of state, expressed in terms of reduced
variables, are explicit functions of the reduced temperature, the packing fraction, the number of
monomers segments forming the chain, and the parameter l which characterises the range of the the
attractive potential. The relevance of the last parameter in the application of the theory to n-alkanes
and n-perfluoroalkanes is explicitly shown with the SW expressions. An accurate description of the
vapour pressure and the saturated liquid densities is obtained, with a simple dependence of the
parameters of the monomer potential on the number of carbons. The extension of our SAFT-VR
expressions to mixtures is also presented in terms of a simple expression for the mean-attractive
energy for mixtures, based on a straightforward generalisation of the theory for pure components.
© 1997 American Institute of Physics. @S0021-9606~97!50306-0#
I. INTRODUCTION
The development of accurate equations of state firmly
based in statistical mechanics is one of the main goals in
applied physical science since it allows for an accurate description of the thermodynamic properties of real substances.
The first molecular theory of fluids is due to van der Waals
~VDW!.1 The Helmholtz free energy A of the van der Waals
equation of state ~EOS! is given by2
A IDEAL A HS A VDW
A
5
1
1
NkT
NkT
NkT
NkT
5ln~ r L 3 ! 212ln~ 124 h ! 2
a VDWr
,¬
kT
~1!
where T is the temperature, k is the Boltzmann constant, N
the total number of molecules of volume b, r 5N/V is the
number density ~with V the volume!, h 5 r b is the packing
fraction, L is the de Broglie wavelength, and a VDW is the
attractive VDW constant. The essence of the EOS is the
inclusion of a contribution A HS due to the hard-sphere ~HS!
4168¬
J. Chem. Phys. 106 (10), 8 March 1997¬
repulsive interactions between the molecular cores, and a
contribution A VDW due to the dispersive van der Waals attractive interactions. In the original expression, van der
Waals approximated the hard-sphere free volume in terms of
the pair contributions. If the more accurate Carnahan and
Starling3 hard-sphere expression is used to describe the repulsive contribution a simple augmented version of the van
der Waals EOS is obtained4,5
A
4 h 23 h 2 a VDWr
5ln~ r L 3 ! 211
.¬
2
NkT
kT
~ 12 h ! 2
~2!
By introducing two basic molecular features such as the
size and attractive interactions, the augmented van der Waals
EOS gives a very good first approximation to the properties
of real substances. Several perturbation theories, which go
beyond the van der Waals description, have been applied to
the statistical mechanics of liquids,6–10 and have improved
the description of the liquid phase behaviour by employing
effective hard-sphere systems to describe the repulsive and
attractive interactions. With the knowledge gained from
0021-9606/97/106(10)/4168/19/$10.00¬
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© 1997 American Institute of Physics
Gil-Villegas et al.: Statistical associating fluid theory
other statistical mechanical methods such as computer simulation, integral equations, functional analysis, etc., it is now
possible to obtain a simple and accurate description of fairly
complex real substances.
Some molecules are, however, highly non-spherical
and/or possess highly directional attractive forces such as
hydrogen bonds. The standard techniques tend to fail in this
case, and new approaches are required. In recent years, a
considerable effort has been made in the development of
theories for fluids of associating chain molecules. The most
successful theories for such systems originate in the seminal
work of Wertheim.11–16 The statistical associating fluid
theory ~SAFT!17,18 is one such theory. The SAFT approach
provides a molecular based EOS that has been used extensively to correlate and predict experimental results for a wide
variety of substances. It is evident from even a brief overview of recent literature that the SAFT equation of state is
becoming one of the most accurate and versatile tools in the
description of fluid phase equilibria, while still retaining a
relatively simple van der Waals-like form. The SAFT approach has been used, with great success, to correlate the
data for the phase equilibria of over 100 pure components,19
and 60 binary mixtures.20 Yu and Chen21 have also used
SAFT to examine the liquid-liquid phase equilibria for 41
binary and 8 ternary mixtures, and include a number of useful correlations for the parameters. The liquid-liquid equilibria in ternary mixtures can be predicted with SAFT from the
corresponding binary data.22 Other systems which have been
examined to date with the SAFT approach include carbon
dioxide mixtures with bitumen23 and methylnaphthalene,24
polynuclear aromatics in ethene and ethane,25 binary and ternary systems of n-alkanes containing ethene and 1-butene,26
methanol¬ 1 ethene¬ mixtures,27 systems¬ containing
alcohols,28 near- and super-critical water,29 and carbon dioxide mixtures with acetonitrile and acrylic acid.30 SAFT generally provides an excellent description of the thermodynamic properties of these mixtures. The approach is also
being used to correlate and predict the phase behaviour of a
wide variety of complex polymer systems ~e.g., see Ref. 31!.
The general form of the SAFT Helmholtz free energy for
associating chain molecules is given by
A IDEAL A MONO. A CHAIN A ASSOC.
A
5
1
1
1
.¬
NkT
NkT
NkT
NkT
NkT
~3!
In this equation A IDEAL is the ideal free energy, A MONO. is the
excess free energy due to the monomer segments, A CHAIN is
the contribution due to the formation of the chains of monomers, and A ASSOC. is the term that describes the contribution
to the free energy due to intermolecular association. In the
original¬ SAFT¬ EOS¬ for¬ chains¬ of¬ Lennard-Jones
segments,17,18 a perturbation expansion is used to describe
the monomer contribution and the hard-sphere radial distribution function is used to describe the chain contribution.
The simplest version of the theory involves associating
chains of hard-sphere segments where the attractive interactions are treated at the mean-field van der Waals level.32 We
shall refer to the this augmented van der Waals EOS for
chain molecules as the SAFT-HS approach. This simplified
4169
SAFT-HS approach gives a good description of the phase
equilibria of a number of systems including the critical behaviour of n-alkanes,33 the upper critical solution temperatures of mixtures of alkanes and perfluoroalkanes,34 the highpressure critical lines of mixtures of alkanes and water,35 and
mixtures containing hydrogen fluoride.36 SAFT-HS is seen to
work best in systems with strong association, where the dispersion forces can be adequately represented as a weak
mean-field background interaction.
The SAFT expressions are continually being improved.
An accurate representation of the monomer-monomer distribution function has been included to deal with chains of
Lennard-Jones ~LJ! segments,37–40 and the approach has
been extended to different types of monomer segments such
as square wells ~SW!.41 The effect of many-body interactions
in the chain has also been included in dimer versions of the
theory for chains formed from both LJ42 and SW43 chains. A
SAFT EOS with the simplified SW dispersion term used in
the simplified perturbed hard chain theory ~SPHCT! has been
proposed,44 but it should be noted that this description of the
attractive interactions has been brought into question.45 As
far as molecular association is concerned only the possibility
of chain and tree-like aggregates are considered in the original SAFT ~and Wertheim! approach. The possibility of
double¬ bonding,46 ring¬ formation,47–50 and¬ bond
co-operativity51 have recently been examined. The effect of
arbitrary monomer segments with attractive potentials of
variable range ~VR! have not been examined to date.
In this paper we report a general version of SAFT for
chain molecules formed from hard-core monomers with an
arbitrary potential of variable range ~VR!. The approach is
then extended to soft-core potentials and to mixtures beyond
the van der Waals one-fluid level. The SAFT-VR theory developed here improves the chain contribution and the meanfield van der Waals description for the dispersion forces of
the SAFT-HS treatment. It also broadens the scope of the
original SAFT EOS by providing an additional parameter
which characterises the range of the attractive part of the
monomer-monomer potential. This variable-range EOS gives
an excellent description of the properties of a wide variety of
systems where non-conformal effects may be important. In
this first paper we exemplify this with an application to
vapour-liquid properties of pure alkanes and perfluoroalkanes, where the ranges of the monomer-monomer potentials
turn out to be quite different.
Before we embark on a description of the SAFT theory it
is important to acknowledge other approaches for associating
chain molecules ~see Ref. 35 for a short overview!. The Wilson, NRTL and UNIQUAC52 activity coefficient models are
commonly used to correlate data, but are of little predictive
value. The perturbed hard chain theory ~PHCT!53 and its
simplified version SPHCT54 have been used together with
the¬ associating¬ perturbed¬ anisotropic¬ chain¬ theory
~APACT!55 to deal with associating chain molecules. Since
the SAFT approach is firmly cast at a molecular level with
well defined parameters it has distinct advantages over these
other approaches. As a direct consequence, the SAFT expres-
J. Chem. Phys., Vol. 106, No. 10, 8 March 1997
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Gil-Villegas et al.: Statistical associating fluid theory
4170¬
sions can be tested directly with computer simulation data,
allowing improvement to be made.
II. INTERMOLECULAR POTENTIAL MODELS
The model molecules that we are going to consider in
this work consist of a chain of attractive spherical monomeric ~M! segments of diameter s interacting through the
intermolecular potential
u M ~ r ! 5u HS~ r; s ! 2 ef ~ r;l ! .
~4!
The monomer-monomer potential consists of a hard-sphere
repulsive interaction u HS, defined by
u HS~ r; s ! 5
H
`
if¬ r, s
0
if
r. s ,
~5!
and an attractive interaction of depth 2 e and shape
f (r;l), where l is a parameter associated with the range of
the attractive forces. The generic potential given by Eq. ~4!
depends on three parameters: s and e are the usual length
and energy parameters that define the corresponding states
behaviour, and the range l allows the treatment of nonconformal properties for the fluid under consideration. By
changing l, the shape of the attractive well is modified, and
the thermodynamic effect of this is that two systems of particles interacting through potentials like ~4! but differing in
l will not obey corresponding states. The types of hard-core
potentials that are considered in this paper include the
square-well ~SW!, Sutherland ~S!, and Yukawa ~Y! models
although the theoretical treatment is completely general.
The shape of the square-well is described by
f SW~ r;l ! 5
H
1
if
s ,r,l s
0
if
r.l s ,
~6!
so that the energy is constant over the range of interaction.
The significance of l as the range of interaction is clear in
this case.
The generalised Sutherland potential is described by a
power law in l as
f S ~ r;l ! 5 ~ s /r ! l .¬
~7!
The parameter l controls the decay of interaction, and hence
the range of the potential. It should not be confused with the
range itself as in the case of the square well. The Sutherland
potential is particularly useful in the treatment of multipolar
interactions, and allows one to make contact with the softcore potentials such as the Lennard-Jones ~LJ! model.
The Yukawa potential is given by
f ~ r;l ! 5
Y
e
2l ~ r/ s 21 !
r/ s
.¬
~8!
By convention the range of the attractive forces in this model
is characterised by l 21 . The Yukawa model finds a particular use in the description of screened Coulombic interactions
found in electrolytes and colloids.
Although all of these potentials are simplifications of the
true interactions between molecules, they allow a highly accurate description of the properties of real substances. An
extra advantage is that the statistical mechanics of hard-core
attractive models is very well known, and for some cases
there are analytical results; these analytical solutions provide
a useful guide for the development of an accurate EOS.56–70
III. GENERAL SAFT APPROACH FOR PURE FLUIDS
Following Wertheim’s perturbation theory for associating fluids, the Helmholtz free energy A for N chains formed
from m monomeric segments can be written in the form
given by Eq. ~3!. For a pure component fluid of associating
chain molecules the different contributions to the free energy
are given as follows.
A. Ideal contribution
The free energy of an ideal gas is given by2
A IDEAL
5ln~ r L 3 ! 21,¬
NkT
~9!
where it is important to note that r is the number density of
chain molecules and not of monomer segments. Since the
ideal term is treated separately, the other terms are all excess
free energies.
B. Monomer contribution
The contribution to the free energy due to the monomers
(m of which make up each chain! is
A MONO.
AM
5m
5ma M ,¬
NkT
N s kT
~10!
where N s is the total number of spherical monomers, and
a M 5A M /(N s kT) is the excess Helmholtz free energy per
monomer.
At the level of the mean-field approximation the monomer free energy can be described in terms of the augmented
van der Waals expression ~2! as
a M 5a HS2
a VDWr s
,¬
kT
~11!
where a HS5A HS/(N s kT) is obtained from the Carnahan and
Starling3 equation, and r s 5N s /V is the number density of
monomer segments. Note that r s 5 r m. The van der Waals
attractive constant is obtained from the monomer-monomer
interaction as
a VDW52 p e
E
`
s
r 2 f ~ r ! dr
E
eE
52 ps 3 e
53b VDW
`
1
x 2 f ~ x ! dx
`
1
x 2 f ~ x ! dx,¬
~12!
where b VDW54b is the van der Waals size parameter, corresponding to the volume excluded by two spherical particles
of volume b5 ps 3 /6, and x5r/ s . We will refer to the van
der Waals mean-field approximation as the mean-field limit.
The augmented van der Waals approximation is used to de-
J. Chem. Phys., Vol. 106, No. 10, 8 March 1997
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Gil-Villegas et al.: Statistical associating fluid theory
scribe the monomer free energy in the SAFT-HS
approach.32,35 In order to improve the accuracy in the representation of experimental data it is necessary to use a better
description of the dispersion forces. This can be done by
using a high-temperature perturbation expansion for the free
energy a M which will be described in Section IV A ~see also
Ref. 18!.
C. Chain contribution
The contribution to the free energy due to the formation
of a chain of m monomers is37
A CHAIN
52 ~ m21 ! lny M ~ s ! ,
NkT
~13!
where y M ( s ) is the monomer-monomer background correlation function evaluated at hard-core contact; if g(r) is the
radial distribution function, then y(r)5exp@uM(r)/kT#g(r).
A simple approximation for y M ( s ) is
y M ~ s ! 5y HS~ s ! ,
~14!
since the structure of the hard-sphere reference is well
known, and the structure of a fluid is determined principally
by the repulsive interactions; analytical expressions for the
contact value of the radial distribution function g HS( s ) are
available. The hard-sphere contact value is used in the original SAFT and SAFT-HS expressions. The monomermonomer structure for Lennard-Jones37–40 and square-well
~with range l51.5) 41 segments have also been used. In the
SAFT-VR theory we use a perturbation expansion for the
contact value of the monomer-monomer radial distribution
function g M ( s ) for general hard-core potentials of variable
range ~see Section IV B!.
D. Association contribution
The contribution due to association for s sites on chain
molecules is obtained from the theory of Wertheim as32
A ASSOC.
5
NkT
F( S
s
a51
ln X a 2
D G
Xa
s
1 ,¬
2
2
~15!
where the sum is over all s sites a on a molecule, and X a is
the fraction of molecules not bonded at site a. The latter
quantity is obtained by a solution of the following mass action equation:
X a5
1
11 ( sb51 r X b D a,b
.¬
~16!
The function D a,b characterises the association between site
a and site b on different molecules. It can be written in terms
of the contact value g M ( s ) of the monomer-monomer
radial¬ distribution¬ function,¬ the¬ Mayer¬ function
f a,b 5exp(2ca,b /kT)21 of the a-b site-site bonding interaction c a,b , and the volume K a,b available for bonding as71
D a,b 5K a,b f a,b g M ~ s ! .
~17!
The bonding volume K a,b can be determined from the parameters of the bonding site such as its position and range.71
4171
As for the chain contribution, the original SAFT and the
SAFT-HS approaches approximate g M ( s ) by g HS( s ).
From these equations it follows that the free energy of
the associating chain molecules can be obtained from a
knowledge of the properties of the monomer: the free energy
per monomer a M , and the contact value of the background
correlation function y M . The specific expressions for a M and
y M used in our SAFT-VR approach are presented in the following sections.
IV. SAFT-VR APPROACH FOR PURE FLUIDS
A. Monomer contribution
The standard statistical mechanical theory for hard-core
systems is the Barker and Henderson perturbation theory
~PT!.6,7,72 The theory originates in the high-temperature expansion ~HTE! approach of Zwanzig,73 in which one uses a
hard-sphere system as a reference fluid and assumes that the
attractive term ef (r;l) acts as a perturbation. This enables
us to express the monomer Helmholtz free energy as a series
expansion in the inverse of the temperature b 51/kT 72
a M 5a HS 1 b a 1 1 b 2 a 2 1 •••,¬
~18!
where a 1 and a 2 are the first two perturbation terms associated with the attractive energy 2 ef .
The mean-attractive energy a 1 is given by72
a 1 522 pr s e
E
`
s
523 r s b VDWe
r 2 f ~ r ! g HS~ r ! dr
E
`
1
x 2 f ~ x ! g HS~ x ! dx.¬
~19!
This corresponds to the average of the monomer-monomer
potential energy calculated with the hard-sphere structure; at
first order in expansion ~18!, the structure of the monomer
fluid is the same as that of the reference hard-sphere system.
The van der Waals mean-field energy,
52 r s a VDW,¬
a VDW
1
~20!
is obtained from Eq. ~19! by taking g HS(r)51 for all intermolecular distances. This corresponds to the assumption of a
random correlation between the position of the particles. The
second order term a 2 describes the fluctuation of the attractive energy as a consequence of the compression of the fluid
due to the action of the attractive well. The exact determination of this term requires a knowledge of higher-order correlation functions. Since, in practice, there is little information
about these functions, an approximation must be used for
a 2 . Barker and Henderson6,7,72 proposed approximate expressions for a 2 based on the idea that the energy fluctuations
described by a 2 are correlated to fluctuations of the number
of particles inside the attractive well. These fluctuations are
related to the compression of the fluid, an effect that can be
taken into account at the macroscopic-thermodynamic level
or, more realistically, at the local level, by noting that the
compression is described by the variation of the local density
of the fluid:
J. Chem. Phys., Vol. 106, No. 10, 8 March 1997
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Gil-Villegas et al.: Statistical associating fluid theory
4172¬
a 2 52 pr s e 2 kT
E
52 pr s e 2 K HS
`
0
r 2 @ f ~ r !# 2
]
]r s
E
`
s
]r s g HS~ r !
dr
] P HS
In this factorisation the integral is basically the van der
Waals attractive constant, so that we can rewrite Eqs. ~19!
and ~24! as
r 2 @ f ~ r !# 2 r s g HS~ r ! dr
H
3 23 r s b VDWe
E
`
1
J
x 2 @ f ~ x !# 2 g HS~ x ! dx ,¬
~21!
where P and K 5kT( ]r s / ] P HS) T are the pressure and
the isothermal compressibility of the hard-sphere fluid. The
original choice for the hard-sphere isothermal compressibility was the Percus-Yevick expression,72
HS
K HS5
HS
~ 12 h ! 4
.¬
114 h 14 h 2
~22!
Since the bracketed term in Eq. ~21! corresponds to the
mean-attractive energy for an effective potential f 2 , we can
rewrite the local compressibility approximation ~LCA! for
a 2 as
a 2 5 21 e K HSr s
]a*
1
,¬
]r s
~23!
where
VDW
e
a*
1 523 r s b
E
`
1
x 2 @ f ~ x !# 2 g HS~ x ! dx.¬
`
1
b
a
x 2 C ~ x ! g HS~ x ! dx,¬
~25!
E
1
where j 1 , j 2 P @ 1,` # and the van der Waals attractive parameters are given by Eq. ~12!, calculated with f or f 2 . The
distances j i in expressions ~28! and ~29! depend on the density and range, but this dependence is very smooth.76 The
mean-attractive energy a 1 for the square-well system has
been analysed using molecular dynamics simulation values
for a 1 in Eq. ~28! together with an analytical expression for
g HS(x) due to Boublı́k,80 to give a simple parameterisation
for j .78 Although the square-well EOS obtained with this
approach is more compact and often more accurate than
other representations reported in the literature, its main limitation is that it can not easily be extended to mixtures. An
alternative approach is used here, based on the fact that for
different monomer potentials, Eqs. ~6!–~8!, j is almost constant and close to the contact value of 1 over a wide range of
densities for different ranges of the potential. For example,
for SW monomers j '1.05 for l51.1, j '1.25 for l51.5,
and j '1.45 for l51.8. A similar behaviour is observed for
other models. We can thus expect that the Taylor expansion
of g HS around the contact value x51,
1
S D
] 2 g HS
]x2
S D
] g HS
]x
~ j 21 !
x51
~ j 21 ! 2 1 . . . , ¬
~30!
x51
should be very convergent. Moreover, since the leading term
in Eq. ~30! is the contact value g HS(1; h ), it is reasonable to
represent the full function g HS( j ; h ) by its contact value, but
evaluated at an effective packing fraction h eff such that
g HS~ j ; h ! 5g HS~ 1; h eff! .¬
~31!
HS
f ~ x ! h ~ x ! dx5 f ~ j !
E
b
a
~26!
h ~ x ! dx.¬
Since y HS5g HS in @ 1,` # we can apply Eq. ~26! to Eq. ~25!
with f (x)5g HS and h(x)5x 2 C(x) to give
`
~29!
g HS~ j ; h ! 5g HS~ 1; h ! 1
where C is an arbitrary potential (C5 f for a 1 , and
C5 f 2 for a *
1 ). The use of the mean-value theorem ~MVT!
from the theory of calculus for the evaluation of integrals
such as Eq. ~25! has been proposed.76–78 In its general
version,79 the MVT states that if f (x) and h(x) are continuous functions in the interval I5 @ a,b # , and h(x).0, then
there is a value j P @ a,b # such that
E
VDW
* g HS~ j 2 ! ,¬
a*
1 52 r s a
~24!
Real systems were accurately described with the LCA in
previous studies,68,74,75 and we have used this approximation
in our work. Alternatively, a simpler but physically less realistic approach based on the macroscopic compressibility
could be used, where a 2 5K HSa 1 /2.72
From a mathematical point of view, it follows that the
application of the high-temperature expansion theory reduces
to the calculation of a single type of integral,
E
~28!
and
]
5 eK rs
]r s
HS
1
2
a 1 52 r s a VDWg HS~ j 1 !
x C ~ x ! g ~ x ! dx5g ~ j !
2
HS
HS
E
`
1
x C ~ x ! dx.¬
2
~27!
This is justified since g (x) is a decreasing function in
the interval of values of the mean-value theorem distances
j , even for long-range interactions and for the whole range
of liquid densities. It is always possible to find an equivalent
hard-sphere system with a smaller packing fraction h eff that
satisfies Eq. ~31!. As an example we show g HS(x) for a highdensity fluid ( h 50.36) in Fig. 1; the corresponding meanvalue theorem distance for a square-well system with
l51.5 is j 51.24, and the value of g HS( j ) is the same as the
contact value g HS(1) for h eff50.11.
Using the Carnahan and Starling3 EOS contact value we
get a very compact expression for the mean-attractive energy
as
g HS~ 1; h eff! ,¬
a 1 52 r s a VDWg HS~ 1; h eff! 5a VDW
1
where
J. Chem. Phys., Vol. 106, No. 10, 8 March 1997
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~32!
Gil-Villegas et al.: Statistical associating fluid theory
FIG. 1. The relationship between g HS( j ; h ) and g HS(1; h eff). The hardsphere pair radial distribution function g HS(x) obtained from the solution of
the Ornstein-Zernike equation with the Malijevsky and Lablı́k ~Ref. 81!
bridge function for h 50.36 is shown as the solid curve. The value of
g HS( j ; h ) with j 51.24 ~dotted curve! corresponds to the contact value
g HS(1,h eff) where h eff50.11 ~dashed curve!.
g HS~ 1; h eff! 5
12 h eff/2
,¬
~ 12 h eff! 3
~33!
with a similar expression for a *
1 . It is now necessary to
obtain the dependence of the equivalent packing fraction
h eff on the actual value of h and on the range of the potential. This can be achieved by using exact values of a 1 obtained directly from computer simulations or by integrating
Eq. ~19! using an accurate representation for g HS(r). Here,
we follow the latter method, and solve the Ornstein-Zernike
equation with the Malijevsky and Labı́k81 formula for the
hard-sphere bridge function. The g HS(r) values obtained in
this way are as accurate as computer simulation results. We
apply this approach to several hard-core systems. The near
exact values of the mean-attractive energy are used to solve
for the appropriate h eff over a range of densities and ranges
of the potential, and its behavior is then described by a polynomial parameterisation. The specific expressions for the
various potential models are now described.
1. Square-well fluids
The square-well fluid is the simplest case of a hard-core
system with an attractive interaction defined by Eq. ~6!. The
square-well mean-attractive energy for ranges 1.1<l<1.8 is
given by
VDW HS
a SW
g ~ 1; h eff! ,¬
1 5a 1
~34!
where in this case the van der Waals attractive parameter is
524 h e ~ l 3 21 ! .¬
a VDW
1
~35!
4173
FIG. 2. Vapour-liquid coexistence densities for square-well monomer fluids
with different values of the potential range l compared with the Gibbs
ensemble simulation data points of Vega et al. ~Ref. 84!. The solid curves
correspond the SAFT-VR approach, and the dashed curves to the EOS of
Chang and Sandler ~Ref. 68!. The curves are labelled with the different
values of the range l.
The following parameterisation for h eff( h ,l) is obtained:
h eff5c 1 h 1c 2 h 2 1c 3 h 3 ,¬
SDS
~36!
where the coefficients c n are given by the matrix
c1
c2
5
c3
2.25855¬
21.50349¬ 0.249434
20.669270¬ 1.40049¬ 20.827739
10.1576¬
SD
215.0427¬
5.30827
D
1
3
l
l2
.¬
~37!
SW
for this model, the fluctuation term a 2 is
Since a *
1 5a 1
given directly by the first density derivative of a SW
1 ,
1
HS
a SW
2 5 2 eK h
] a SW
1
.¬
]h
~38!
SW
By including these expressions for a SW
1 and a 2 in Eq. ~18!
and then in Eq. ~10! we obtain the SAFT-VR EOS for
square-well fluids of variable range.
Phase equilibria in a pure fluid require that the temperature, pressure, and chemical potential of the two coexisting
phases are equal.82 The chemical potential ( m ) and pressure
can be obtained from the Helmholtz free energy using the
standard thermodynamic relations: m 5( ] A/ ] N) T,V and
Z5 PV/(NkT)5 m /(kT)2A/(NkT). We solve the conditions for phase equilibria of the square-well system numerically using the simplex method.83 In Fig. 2 we show the
J. Chem. Phys., Vol. 106, No. 10, 8 March 1997
Copyright ©2001. All Rights Reserved.
Gil-Villegas et al.: Statistical associating fluid theory
4174¬
prediction of the vapour-liquid coexistence curve given by
SAFT-VR, and compare the predictions with existing Gibbs
ensemble simulation data.84 The SAFT-VR EOS provides an
accurate representation of the simulation data for several values of the range l. Although a more accurate representation
for a 2 than the local compressibility approximation can be
used, e.g., empirical fits to simulation results,64,78 the approximation is a simple recipe for a 2 with a firm theoretical
footing. Other theoretical descriptions for a 2 have been
reported,63,67 but the expressions are considerably more complex, and in the end make little difference to the predictive
capabilities for real systems. The coexistence curves obtained with the analytical solution of Chang and Sandler68
using the Percus-Yevick approximation and LCA are also
included in Fig. 2. Our SAFT-VR approach provides a description which is of comparable accuracy, yet retains much
more manageable expressions for a 1 and, as we will show
later, for y M ( s ). This is of prime importance in extending
the SAFT-VR treatment to mixtures.
SDS
c1
c2 5
20.943973¬ 0.422543¬ 20.0371763¬
2. Sutherland fluids
The attractive well for the Sutherland potential, given by
Eq. ~7!, has a more realistic shape than the discontinuous
square well. By considering a variable power l, which characterises the range of the potential, we can model different
angle-averaged multipolar forces. For powers in the range
3,l<12, we have obtained the expressions for a 1 and a 2 .
The mean-attractive energy is given by
g HS~ 1; h eff! ,¬
a S1 5a VDW
1
where
a VDW
524 h e
1
3l
,¬
l23
~40!
h eff~ h ,l ! 5c 1 h 1c 2 h 2 ,¬
~41!
with
0.00116901
D
SD
1
l
~42!
l 2 .¬
l
3
5212h e ~ l 21 1l 22 ! ,¬
a VDW
1
For the Sutherland potential
S
a*
1 ~ l ! 5a 1 ~ 2l ! ,¬
~43!
~46!
and the parameterisation for h eff( h ,l) is
h eff~ h ,l ! 5c 1 h 1c 2 h 2 ,¬
so that
] a S1 ~ 2l !
]h
.¬
~44!
This means that in order to calculate a S2 we simply need to
evaluate a S1 but for a potential of range 2l. It is important to
recognise that this is strictly valid only for l,6, but for
l.6, a S2 tends to smaller values with increasing range and
becomes negligible relative to a S1 . The SAFT-VR EOS for
Sutherland fluids of variable range is obtained by including
these expressions for a S1 and a S2 in Eq. ~18! and then in Eq.
~10!. These expressions are useful not only when considering
dispersion forces but also for the evaluation of some free
energy terms in perturbation theories of point multipolar
fluids.85
SDS
c1
a Y1 5a VDW
g HS~ 1; h eff! ,¬
1
0.900678¬ 21.50051¬
~45!
0.776577
c 2 5 20.314300¬ 0.257101¬ 20.0431566
SD
D
1
3
l 21
l 22
~48!
.¬
The fluctuation term is obtained from
VDW
* g HS~ 1,h * ! ,
a*
1 ~ l ! 5a 1
~49!
where
* 526 h e l 21 ,¬
a VDW
1
The attractive well for a Yukawa potential is described
by Eq. ~8!; in this case l 21 describes the range of the potential. The expressions for a 1 , a *
1 and a 2 for inverse ranges of
1.1<l<4 can be expressed as follows:
~47!
with
3. Yukawa fluids
where
S D
and in this case the parameterisation for h eff( h ,l) is
0.370942¬ 20.173333¬ 0.0175599¬ 20.000572729
a S2 ~ l ! 5 21 e K HSh
~39!
~50!
and
g HS~ 1,h * ! 5
12 h * /2
,¬
~51!
h * ~ h ,l ! 5d 1 h 1d 2 h 2 ,¬
~52!
~ 12 h * ! 3
with the parameterisation
J. Chem. Phys., Vol. 106, No. 10, 8 March 1997
Copyright ©2001. All Rights Reserved.
Gil-Villegas et al.: Statistical associating fluid theory
where
SDS
d1
SD
4175
1
0.989601¬
20.872203¬ 0.320808¬
0.0¬
0.0
d2 5 20.0119152¬ 21.24029¬ 2.41636¬ 22.01922¬ 0.647565
D
l 21
l 22
l 23
~53!
.¬
l 24
The final expression for a 2 is
a Y2 ~ l ! 5 21 e K HSh
]a*
1 ~l!
.¬
]h
~54!
Thus for Yukawa fluid of variable range, the SAFT-VR EOS
is obtained by using a Y1 and a Y2 in Eq. ~18! and then in Eq.
~10!. Although we have used independent parameterisations
for a 1 and a *
1 , the thermodynamics of the Yukawa fluid can
be recast in terms of a *
1 only. By using the properties of
Laplace transforms, it follows that
a 15
]a*
1
2a *
1 .¬
]l
~55!
This way of obtaining a 1 , however, makes the application of
the Yukawa model to mixtures more complex.
As for the square-well fluids we have numerically determined the phase equilibria of the Yukawa fluids with the
SAFT-VR EOS for two values of the attractive range parameter; the vapour-liquid coexistence curves are presented in
Fig. 3 together with Gibbs ensemble simulation data.86 It is
clear from this comparison that the SAFT-VR approach provides a good description of the fluid, apart from the critical
region where the critical temperature is overestimated. We
also examine the effect of including just the first perturbation
term a 1 , and both terms a 1 and a 2 . The critical temperature
is lowered by inclusion of the a 2 term as are the values of the
coexisting liquid densities. This behaviour is also seen for
the square-well and Sutherland fluids. It should be noted,
however, that the effect is weaker than for the square-well
case, a consequence of the faster convergence of the hightemperature expansion for the continuous potentials. The
prescription of the mean-spherical solution according to the
high-temperature expansion proposed recently by Henderson
et al.,69 at the first-order level, is also shown in Fig. 3. At this
level of approximation the SAFT-VR approach is seen to be
as accurate as this analytical solution, even for values of l as
high as 10.
4. Soft repulsive fluids
Although our SAFT-VR approach does not directly describe intermolecular potentials with a soft repulsive cores, it
can be extended to deal with such systems. Specifically, we
examine the Mie m2n potentials,87 of which the LennardJones (m56 and n512) is the most common example
u M 5C e
FS D S D G
s
r
n
2
s
r
m
,¬
~56!
where
SD
n
n
C5
n2m m
FIG. 3. Vapour-liquid coexistence densities for Yukawa monomer fluids
with different values of the potential range l 21 compared with the Gibbs
ensemble simulation data points of Lomba and Almarza ~Ref. 86!. The solid
curves correspond to the SAFT-VR approach up to first order (a 1 only!, and
the dashed curves up to second order (a 1 and a 2 ). The curves are labeled
with the different values of l. For the case of l51.8 the theory is also
compared with the first order mean-spherical EOS due to Henderson et al.
~Ref. 69! ~dotted curve!.
m
n2m
.¬
~57!
According to the Barker and Henderson perturbation
theory, the free energy can still be expressed as the hightemperature expansion ~18! with the first and second order
terms a 1 and a 2 defined by Eqs. ~19! and ~21!, but evaluated
with an effective packing fraction
3
h BH~ T ! 5 r s ps BH
/6.¬
The effective hard-sphere diameter,
J. Chem. Phys., Vol. 106, No. 10, 8 March 1997
Copyright ©2001. All Rights Reserved.
~58!
Gil-Villegas et al.: Statistical associating fluid theory
4176¬
s BH~ T ! 5
E
s
0
~ 12exp~ 2 b u M !! dr,¬
~59!
is now a function of temperature. Here, s defines the position where u M changes sign. From a formal point of view the
only difference between the hard- and soft-core systems is
the use of a temperature-dependent packing fraction in the
evaluation of the thermodynamic properties, and the factorisation of the mean-attractive energy a 1 is still valid,
a 1 52 a VDWr s g HS@ h eff~ h BH~ T !!# ,¬
~60!
where h eff depends on the temperature through h BH(T). The
fluctuation term a 2 is given by Eqs. ~23! and ~24!.
We consider the Mie potentials since this family of potentials represents a sum of a repulsive Sutherland potential
of range n and an attractive Sutherland potential of range
m. The mean-attractive energy can thus be expressed as the
sum of two Sutherland a 1 terms as
S
S
a MIE
1 5C @ 2a 1 ~ h BH ;l5n ! 1a 1 ~ h BH ;l5m !# ,¬
~61!
a S1
where
corresponds to the mean-attractive energy for a
Sutherland system with exponent l. The second-order term
is obtained in similar way as
S
S
a MIE
2 ~ l ! 5C @ 2a 2 ~ h BH ;l5n ! 1a 2 ~ h BH ;l5m !#
5
FIG. 4. Vapour-liquid coexistence densities for a Lennard-Jones fluid compared with the Gibbs ensemble simulation data points of Panagiotopoulos
~Ref. 88!. The solid curve corresponds to the EOS of Johnson et al. ~Ref.
89!. The prediction obtained with the SAFT-VR approach up to first order is
shown as a dotted curve and up to second order is shown as a dashed curve.
]
C
e K HSh BH
@ 2a S1 ~ 2n ! 1a S1 ~ 2m !# .
2
] h BH
~62!
Furthermore, since a S2 is very small for large values of l, and
since n.m, we can simplify the calculation and approximate
a MIE
as
2
S
a MIE
2 'C a 2 ~ h BH ;l5m ! .¬
~63!
Here, we have assumed that only the attractive part of the
potential u M contributes to the fluctuation of the total energy.
Once the diameter s BH has been evaluated for each set of
values of m and n, the SAFT-VR equation of state for the
Sutherland family obtained in Section IV A 2 can be used to
describe the thermodynamic properties of the soft-core system.
As an example, we examine the Lennard-Jones fluid
with m56 and n512. In this case we parameterise the diameter using a cubic equation in T * 5kT/ e :
s BH
50.99543820.0259917T *
s
10.00392254T * 2 20.000289398T * 3 .¬
~64!
The Lennard-Jones vapour-liquid coexistence curve, numerically determined with the SAFT-VR approach, is shown in
Fig. 4 to first (a 1 ) and second (a 1 and a 2 ) order in the
perturbation expansion. The Gibbs ensemble simulation
data88 and the accurate equation of state of Johnson et al.,89
obtained by fitting to simulation data, are also shown for
comparison. The SAFT-VR equation of state for LennardJones fluids only requires 12 constants, and yet is of comparable accuracy ~at least for the coexistence curve! as the
Johnson et al. expression which is described by 32 constants.
B. Monomer background correlation function y M at
contact
It is clear from Eqs. ~13! and ~17! that the quantity which
is required for the evaluation of the contributions to the free
energy due to the formation of the chain molecules and due
to intermolecular association is the contact value of the
monomer background correlation function y M ( s ). Of
course, a true perturbation theory of non-spherical molecules
would require the full angular distribution function of the
reference system which is not known for molecules with arbitrary shape. The use of the monomer-monomer distribution
function is natural within a Wertheim type approach, and the
approximation is expected to be very accurate for the contribution due to association ~as the interactions are very short
ranged! and we show that it is also adequate in describing the
dispersion forces of chains of monomeric segments. The
property that will be most sensitive to the use of the monomer distribution function is the contribution due to chain
formation. As for the free energy, a high-temperature expansion can be used for the radial distribution function g(r) 72
g M ~ r ! 5g HS~ r ! 1 be g 1 ~ r ! 1 ~ be ! 2 g 2 ~ r ! 1••• .¬
~65!
The background correlation function for the monomers is
obtained through the relation y M (r)5exp@buM(r)#gM(r). To
zeroth order the structure is given by the hard-sphere cavity
function corresponding to the first-order term in the free energy expansion ~the mean-attractive energy!. This is a particular case of the general rule that a (n21)th-order expansion for the structure corresponds to a nth-order expansion
J. Chem. Phys., Vol. 106, No. 10, 8 March 1997
Copyright ©2001. All Rights Reserved.
Gil-Villegas et al.: Statistical associating fluid theory
for the thermodynamics. Since the free energy is described
by a second-order expansion, we have to consider only the
calculation of the first-order term g 1 .
A closed expression for g 1 ( s ) is derived with a selfconsistent method for the pressure P from the Clausius virial
theorem and from the density derivative of the Helmholtz
free energy. The Clausius theorem for the compressibility
factor Z M 5 PV/(N s kT) of the monomer fluid can be written
as72
Z M 511
2p
r
3 s
E
`
r3
0
] exp~ 2 b u M ! M
y ~ r ! dr
]r
2p
r
3 s
E
`
s1
1
E
~66!
s
r3
1
E
] f HS
g ~ r ! dr1•••,¬
r
1
]r
s
3
~67!
where Z HS is the compressibility factor of the hard-sphere
fluid given by the Carnahan and Starling equation.3 Alternatively, the pressure can be obtained from the volume derivative of the free energy, P52 ] A/ ] V (Z M 5 h ] a M / ] h ), and
the compressibility factor can be written as an expansion in
b as
Z 5Z 1 b Z 1 1 b Z 2 . . . , ¬
M
HS
~68!
2
where
Z 15 h
]a1
,¬
]h
~69!
Z 25 h
]a2
.¬
]h
~70!
and
We can equate the two expressions for the compressibility
factor ~67! and ~68!, up to first order in both, and get the
desired formula for g 1
g 1~ s ! 5
1 ]a1
2
4e ]h
E
`
1
x3
x3
] f SW HS
g ~ x ! dx52l 3 g HS~ l ! .¬
]x
g 1~ s ! 5
~73!
1 ]a1
1l 3 g HS~ l ! .¬
4e ]h
] f HS
g ~ x ! dx.¬
]x
~74!
From the formal definition of the square-well mean-attractive
energy,
E
l
1
~75!
x 2 g HS~ x ! dx,¬
] a SW
]
1
523b VDWe r s
]l
]l
2 ps 3
5Z 1
r s be g 1 ~ s !
3
`
~72!
we can determine g HS(l) by applying the Leibniz theorem
~Ref. 90, formula 3.3.7! to give
HS
2p
1
be r s
3
`
VDW
a SW
ers
1 523b
] exp~ bef ! HS
y ~ r ! dr1•••
]r
`
] f SW
52 d ~ x2l ! ,
]x
With this result, Eq. ~71! is given simply by
2 ps 3
2 ps 3
r s g HS~ s ! 1
r s be g 1 ~ s !
3
3
2p
r
3 s
For the square-well attractive potential f SW we have that
E
By using the high-temperature expansion ~65!, the expression for the compressibility factor becomes
Z M 511
1. Square-well fluids
1
] exp~ bef ! M
y ~ r ! dr.¬
]r
r3
explicit function of the mean-attractive energies a 1 and
a*
1 . The specific expressions depend on the particular model
of attractive-well, and must be determined separately.
so that
2 ps 3
511
r sg M~ s !
3
1
4177
~71!
Although Eq. ~71! involves an integral that requires, in principle, a knowledge of the radial distribution function for the
hard-sphere system as a function of the intermolecular distance x, integration by parts allows us to rewrite g 1 ( s ) as an
E
l
1
x 2 g HS~ x ! dx
523b VDWe r s l 2 g HS~ l ! .¬
~76!
Substituting Eq. ~76! into Eq. ~74! we have that
F
G
1 ] a SW
l ] a SW
1
1
2
,¬
g SW~ s 1 ! 5g HS~ s ! 1 b
4
]h
3h ]l
~77!
where a SW
1 is given by Eq. ~34!. The final expression for the
background monomer-monomer correlation function is
y SW~ s ! 5g SW~ s 1 ! exp~ 2 be ! .
~78!
By including this first-order expression for y SW( s ) in Eqs.
~13! and ~17! for the contributions due to chain formation
and association we obtain the SAFT-VR equation of state for
associating chain molecules formed from square-well segments of variable range.
The phase equilibria of the chains of square-well segments are determined as for the monomer square-well fluid
~see Section IV A 1!. The effect of including the zeroth- and
first-order terms for y SW( s ), as given by Eqs. ~77! and ~78!,
on the vapour-liquid equilibria of a tangent square-well diatomic (m52) with range l51.5 is examined in Fig. 5. We
compare the results of the SAFT-VR theory with existing
Gibbs ensemble Monte Carlo simulation results.91 It is clear
that the first-order term is necessary for a good description of
the thermodynamics; the use of the hard-sphere structure
~zeroth-order expansion! is not fully satisfactory. The inadequacy of the mean-field description of the dispersion interactions ~as in SAFT-HS! is also shown. The results of the
theory for the vapour-liquid coexistence of flexible chains of
m54 and 16 tangent square-well segments with range
J. Chem. Phys., Vol. 106, No. 10, 8 March 1997
Copyright ©2001. All Rights Reserved.
4178¬
Gil-Villegas et al.: Statistical associating fluid theory
2. Sutherland fluids
For the Sutherland potential f S ,
]fS
52lx 2 f S ,¬
]x
x3
~79!
which means that the integral term in Eq. ~71! is the same
integral involved in the calculation of the mean-attractive
energy a S1 :
E
`
x3
1
] f S HS
g ~ x ! dx52l
]x
E
`
1
x 2 f S g HS~ x ! dx.¬
According to Eq. ~71!, we find that
F
G
1 ] a S1
l S
2
a ,¬
g S ~ s 1 ! 5g HS~ s ! 1 b
4
]h 3h 1
~80!
~81!
where
y S ~ s ! 5g S ~ s 1 ! exp~ 2 be ! ,
FIG. 5. The SAFT-HS ~AVDW! and SAFT-VR (y HS and y SW) vapourliquid coexistence densities for square-well diatomics with l51.5 compared
with the Gibbs ensemble simulation results of Yethiraj and Hall ~Ref. 91!.
The curve labelled y HS represents the use of the hard-sphere contact value,
and the curve labelled y SW the square-well contact value.
~82!
and
is given by Eq. ~39!. The SAFT-VR equation of state
for associating chains of Sutherland segments is obtained by
including the contact value y S ( s ) in Eqs. ~13! and ~17! for
the contributions due to chain formation and association.
a S1
3. Yukawa fluids
l51.5 are compared with the results for the diatomic system
and with the Gibbs ensemble data of Escobedo and
de Pablo92 in Fig. 6. The SAFT-VR theory reproduces the
main trends and is in reasonable agreement with the simulation data although its adequacy appears to be reduced for
systems with longer chain lengths. In this case the dimer
versions of the SAFT approach42,43 are recommended.
The contact value of the background correlation function
for the Yukawa potential f Y can be expressed in terms of
a 1 , with the help of the properties of Laplace transforms.
Since
]fY
52x 3 l f Y 2x 2 f Y ,¬
]x
x3
~83!
the integral in Eq. ~71! for the Yukawa potential is given by
E
`
1
x3
E
E
] f Y HS
g ~ x ! dx52l
]x
2
`
1
`
1
x 3 f Y g HS~ x ! dx
x 2 f Y g HS~ x ! dx,¬
~84!
where we can again recognise the mean-attractive integral as
the second term on the right-hand side. This integral is proportional to the Laplace transform L of xg HS(x) 90
E
`
1
x 2 f Y g HS~ x ! dx5
E
`
1
xexp@ 2l ~ x21 !# g HS~ x ! dx
5e l L $ xg HS~ x ! % .¬
~85!
Similarly, the other integral involved in Eq. ~84! can be written as
E
`
1
FIG. 6. Vapour-liquid coexistence densities for square-well chains of m
segments compared with the Gibbs ensemble simulation data points of Escobedo and de Pablo ~Ref. 92! (m54 and 16!, and of Yethiraj and Hall
~Ref. 91! (m52). The curves correspond to the SAFT-VR approach.
x 3 f Y g HS~ x ! dx5e l L $ x 2 g HS~ x ! %
52e l
] L $ xg HS~ x ! %
,¬
]l
~86!
where we have applied the property of Laplace transforms
that90
J. Chem. Phys., Vol. 106, No. 10, 8 March 1997
Copyright ©2001. All Rights Reserved.
Gil-Villegas et al.: Statistical associating fluid theory
L $ x n f ~ x ! % 5 ~ 21 ! n
] nL$ f ~ x !%
.¬
]ln
~87!
With Eqs. ~85! and ~86!, expression ~84! is given simply by
E
`
1
F
G
] f HS
]I
g ~ x ! dx5l 2
1I Y 2I Y ,¬
x3
]x
]l
Y
Y
~88!
where
IY5
E
`
1
x 2 f Y g HS~ x ! dx.¬
F
G
1 ] a Y1
l ] a Y1 ~ 11l ! Y
2
1
a1 ,
g Y ~ s 1 ! 5g HS~ s 1 ! 1 b
4
]h 3h ]l
3h
~90!
where
1
y ~ s ! 5g ~ s ! exp~ 2 be ! ,
Y
A. Ideal contribution
The ideal free energy of a mixture is given by2
A IDEAL
5
NkT
S(
D
n
i51
x i lnr i L 3i 21.¬
The sum is over all species i of the mixture, x i 5N i /N is the
mole fraction, r i 5N i /V the number density, N i the number
of molecules, and L i the thermal de Broglie wavelength of
species i.
B. Monomer contribution
As for the pure components, we use the Barker and
Henderson93 perturbation theory for mixtures with the appropriate hard-sphere mixture as a reference. The monomer free
energy is
A MONO.
5
NkT
~91!
is obtained from Eq. ~45!. This expression for the
and
contact value y Y ( s ) is included in Eqs. ~13! and ~17! for the
contributions due to chain formation and association to give
the SAFT-VR equation of state for associating chains of
Yukawa segments.
5
a Y1
S( D
S( D
x im i
i
AM
N s kT
x i m i a M ,¬
i
V. SAFT-VR APPROACH FOR MIXTURES
The SAFT-VR equation of state for pure associating
chain molecules developed in the previous sections is now
extended to mixtures of n components. The monomer segments of the mixture interact with the same type of molecular potentials as for the pure components. The intermolecular
potential between a monomeric segment of component i and
a monomeric segment of component j is
~92!
where e i j is the strength of the i2 j interaction and f i j is the
attractive perturbation. The chain molecules of component i
are formed from m i monomeric segments, and may also associate with the incorporation of the appropriate site-site interactions.
As for the pure component case, the most important advantage of our SAFT-VR approach for mixtures is that the
MVT can still be applied for the evaluation of the meanattractive energy a 1 of a mixture of monomers, and that we
can use the contact value of the radial distribution function
of a hard-sphere mixture in order to obtain a compact expression similar to Eq. ~28!. The remaining terms of the molecular free energy can be calculated with this expression for
a 1 . The SAFT-VR free energy for a mixture has the same
form as Eq. ~3!, and, as for the pure fluids, each of the contributions will be described in turn.
~94!
where m i is the number of spherical segments of chain i. The
monomer free energy per segment of the mixture
a M 5A M /(N s kT) is obtained from the expansion,
a M 5a HS1 b a 1 1 b 2 a 2 ,¬
HS
um
i j 5u i j 2 e i j f i j , ¬
~93!
~89!
The use of Eqs. ~88! in Eq. ~71! provides us with an expression for the contact value
Y
4179
~95!
where each term is now for a mixture of spherical segments.
The free energy of the reference hard-sphere mixture is
obtained from the expression of Boublı́k94 and Mansoori
et al.95
6
a 5
pr s
HS
FS
z 32
z 23
2z0
D
G
z 32
3 z 1z 2
ln~ 12 z 3 ! 1
1
.
12 z 3 z 3 ~ 12 z 3 ! 2
~96!
In this expression r s 5N s /V is the number density of the
mixture in terms of the number of spherical segments. Note
that r s 5 r ( ( i x i m i ), where r is the total molecular number
density of the mixture. The reduced densities z l are defined
as
p
z l5 r s
6
F(
n
i51
G
x s,i ~ s i ! l ,¬
~97!
where s i the diameter of spherical segments of chain i, and
x s,i is the fraction of segments of type i in the mixture. The
overall packing fraction of the mixture is thus given by z 3 ,
which is equivalent to h in the pure component case.
The mean-attractive term a 1 of the mixture is the sum of
the terms for each type of pair interaction ~92!
n
a 15
n
((
i51 j51
x s,i x s, j a i1j , ¬
where
J. Chem. Phys., Vol. 106, No. 10, 8 March 1997
Copyright ©2001. All Rights Reserved.
~98!
Gil-Villegas et al.: Statistical associating fluid theory
4180¬
a i1j 522 pr s e i j
E
`
si j
523 r s b VDW
eij
ij
C. Chain contribution
r 2i j f i j ~ r i j ! g HS
i j ~ r i j ! dr i j
E
`
1
x 2 f i j ~ x ! g HS
i j ~ x ! dx.¬
~99!
Here, b VDW
52 ps 3i j /3, g HS
ij
i j is the pair radial distribution
function for a mixture of hard sphere, and x5r i j / s i j with
s i j 5( s i 1 s j )/2. Since each term given by Eq. ~99! has the
same form as the mean-attractive energy of the pure component, Eq. ~28!, we can again apply the mean-value theorem
~see Section IV A! to obtain a compact expression in terms
of the contact value of g HS
i j such that
a i1j 52 r s a VDW
g HS
ij
i j ~ ri j5j1 ;z3!
eff
52 r s a VDW
g HS
ij
i j ~ r i j 5 s i j ; z 3 ! ,¬
~100!
53b VDW
e i j * x 2 f i j (x)dx. The contact
where now a VDW
ij
ij
HS
value g i j ( s i j ; z 3 ) is evaluated with the Boublı́k94 expression
as
s is j
z2
1
13
g HS
i j ~ si j !5
s i 1 s j ~ 12 z 3 ! 2
~ 12 z 3 !
S
s is j
12
s i1 s j
D
2
z 22
~ 12 z 3 ! 3
a 25
~101!
.¬
n
(i (j
~102!
x s,i x s, j a i2j . ¬
The terms a i2j for each type of pair interaction are defined as
]a*
1
,¬
]r s
ij
a i2j 5 12 e i j K HSr s
~103!
where the Percus-Yevick expression87
~104!
is used for the compressibility of the reference hard-sphere
mixture. As for the pure components, a i2j can be written in
ij
terms of an effective mean-attractive energy a *
which is
1
related to the square of the monomer-monomer attractive interaction f i j through
E
e E
523 r s b VDW
ij
ij
`
si j
`
1
(
r 2i j f 2i j ~ r i j ! g HS
i j ~ r i j ! dr i j
x 2 f 2i j ~ x ! g HS
i j ~ x ! dx,¬
and can be evaluated using the MVT as was done for
~105!
a i1j
.
~106!
where y iiM ( s i ) is obtained from the high-temperature expansion of the pair radial distribution function g iiM ( s i ).
In general, the contact value of the radial distribution
function for segments of species i and j can be written as
g iMj ~ s i j ! 5g HS
i j ~ s i j ! 1 be i j g 1 ~ s i j !
~107!
with
y iMj ~ s i j ! 5g iMj ~ s i j ! e 2 be i j . ¬
~108!
The term g 1 ( s i j ) is obtained from a self-consistent calculation of the pressure using the Clausius virial theorem,
2p
PV
511
r
N s kT
3 s
3
E
`
0
r 3i j
n
n
(i (j x s,i x s, j
] exp~ 2 b u iMj ! M
y i j ~ r i j ! dr i j , ¬
]rij
~109!
and the thermodynamic relation
ZM5rs
S D
]aM
.¬
]r s
~110!
By equating the expressions for the compressibility factor
given by Eqs. ~109! and ~110! we find that
E
] a i1j
g 1~ s i j ! 5
2
2 ps 3i j e i j ]r s
3
so that
g iMj ~ s i j ! 5g HS
i j ~ s i j ! 1 be i j
2
z 0 ~ 12 z 3 ! 4
K HS5
,¬
z 0 ~ 12 z 3 ! 2 16 z 1 z 2 ~ 12 z 3 ! 19 z 32
a 1* i j 522 pr s s 3i j e i j
n
A CHAIN
52
x i ~ m i 21 ! lny iiM ~ s i ! ,¬
NkT
i51
ZM5
The mapping procedure used in Eq. ~100! requires an explicit equation for z eff
3 in terms of the overall packing fraction
of the mixture z 3 . As a first approximation, we can assume
that the parameterisation obtained for the packing fraction
h eff@ h # of the pure component, e.g., Eq. ~36! for the SW
fluid, can be used for z eff
3 @ z 3 # of the mixture, especially if the
sizes of the monomeric segments are not very different.
For the first fluctuation term a 2 we again use the local
compressibility approximation as applied to mixtures:
n
As for the pure component case, the contribution to the
free energy due to linking the monomeric segments together
to form the chain molecules is given in terms of the contact
value of the pair background distribution function of the
monomers32
E
`
x3
1
F
`
x3
1
] f i j HS
g ~ x ! dx,¬ ~111!
]x ij
] a i1j
2 ps 3i j e i j ]r s
3
G
] f i j HS
g ~ x ! dx .¬
]x ij
~112!
This expression for the contact value of the radial distribution function will also be used to determine the contribution
due to association.
D. Association contribution
The contribution to the free energy due to the association
mediated by the s i sites on chain molecules of species i can
be described within the framework of the theory of Wertheim as32
n
A ASSOC.
5
xi
NkT
i51
(
J. Chem. Phys., Vol. 106, No. 10, 8 March 1997
Copyright ©2001. All Rights Reserved.
F( S
si
a51
lnX a,i 2
D G
X a,i
si
1 .¬
2
2
~113!
Gil-Villegas et al.: Statistical associating fluid theory
The first sum is over the species i, and the second is over all
s i sites a on a molecule of type i. The fraction X a,i of molecules of species i not bonded at site a is obtained by the
numerical solution of the following simultaneous equations:
X a,i 5
1
s
j
11 ( nj51 ( b51
r x j X b, j D a,b,i, j
~114!
,¬
where
D a,b,i, j 5K a,b,i, j f a,b,i, j g iMj ~ s i j ! . ¬
~115!
The parameter D a,b,i, j is specific for each a2b site-site
bonding interaction and incorporates the strength of the association¬ in¬ terms¬ of¬ the¬ Mayer¬ function
f a,b,i, j 5exp(2ca,b,i,j /kT)21 of the SW bonding potential
c a,b,i, j as well as the volume available for bonding
K a,b,i, j .
The application of the SAFT-VR equation of state to
study the phase equilibria of mixtures of associating chain
molecules is straightforward. The main advantage of the factorisation used for the mean-attractive energy is that the relatively simple and compact expressions obtained for the pure
components are preserved in mixtures. As was mentioned
earlier, we have assumed that the relationship between the
effective z eff
3 and the actual z 3 packing fractions of the mixture can be approximated by the one obtained between h eff
and h for the pure fluid ~see Section IV A!. Since the validity
of this assumption depends on the magnitude of the size
difference of the monomeric segments, which in many cases
of interest may be quite small, the approximation should be
adequate. Preliminary results for the phase equilibria of binary mixtures of square-well molecules indicate that the use
of relation ~36! for z eff
3 is a good approximation. In essence,
our SAFT-VR equation of state for mixtures goes beyond the
van der Waals one-fluid level in that it describes all of the
contributions in the free energy ~albeit in an approximate
manner! to second order. One could, of course, apply the
commonly used one-fluid approximation ~with the appropriate mixing rules! for all or part of the free energy to obtain a
hierarchy of expressions for the mixtures.82 In future work
we will present a more detailed analysis of the SAFT-VR
approach for mixtures and examine the relative merits of the
various approximations.
VI. ARBITRARY MONOMER INTERACTION
The square-well interaction has the simplest shape of the
family of potentials described by Eq. ~4!. An accurate description of the system is useful in studies of other arbitrary
systems. For example, from the definition of the meanattractive energy and the Leibniz rule we can obtain, via Eq.
~76!, a recipe for the hard-sphere radial distribution function
for any intermolecular separation. For a pure component we
have
F
g HS~ x ! 5 2
and for mixtures
1
]a1
3
2
2 pr s s e l ] l
G
,¬
l5x
~116!
F
g HS
i j ~ x !5 2
4181
] a i1j
2 pr s s 3i j e i j l 2i j ] l i j
1
G
.¬
~117!
l i j 5x
This means that the pair distribution function at any intermolecular separation x can be determined for the meanattractive energy of a system with a range l5x. Eq. ~117! is
particularly interesting, because it enables us to determine
the full distribution function for mixtures of hard spheres; the
expression has the Boublı́k94 limit for the hard-sphere contact value with the prescription used for the calculation of
a i1j , i.e., Eq. ~99!.
By using Eq. ~116! in the definition of a 1 for an arbitrary
monomer-monomer intermolecular potential u M (r), Eq.
~19!, we find that
a 1M 5
E
` ] a SW~ x !
1
]x
1
f ~ x ! dx.¬
~118!
Since u M →0 when x→`, we have, after an integration by
parts, that
a 1M 5
E ] f]
`
1
~ x ! SW
a 1 ~ x ! dx.¬
x
~119!
Similar expressions can be obtained for a 2M and g 1 :
E]]
]
r
E ] f]
]r
a 2M 5 12 e K HSr s
5 21 e K HS
5
]
]r s
`
a SW
1 ~x!
1
E ] f]
s
f ~ x ! 2 dx
~ x ! 2 SW
a 1 ~ x ! dx
x
`
s
x
1
~ x ! SW
a 2 ~ x ! dx,¬
x
`
1
2
~120!
and
g 1~ s ! 5
5
1 ]a1
2
4e ]h
E ] f]
`
1
E
`
1
x3
] f ~ x ! HS
g ~ x ! dx
]x
~ x ! SW
g 1 ~ s ;x ! dx.
x
~121!
SW
In these identities a SW
1 (x) represents a 1 (l5x), with a corresponding notation for the properties. Similar expressions
follow for the mixture.
Although these results are rather formal and are not useful in directly obtaining the equation of state for an arbitrary
monomer intermolecular potential, they provide an insight
into the convergence of the high-temperature expansion.
Within this framework we can consider that the square-well
systems represent a kind of basis for the properties of the
arbitrary system. It should also be noted that since, in most
of the cases, f is a monotonically decreasing function with
distance, the square-well systems with shorter ranges l will
have a preponderant weight in the integrals ~118!, ~120!, and
~121!. By comparing the free energy of the square-well system,
2 SW
a SW5a HS1 b a SW
1 1 b a 2 ,¬
J. Chem. Phys., Vol. 106, No. 10, 8 March 1997
Copyright ©2001. All Rights Reserved.
~122!
Gil-Villegas et al.: Statistical associating fluid theory
4182¬
TABLE I. Optimised SW intermolecular potential parameters for n-alkanes
obtained by a fit to experimental vapour-pressure and saturated liquid density data from the triple point to the critical point ~Ref. 96!. m is the number
of spherical segments in the model, l the SW range parameter, s the diameter of each segment, and e is the well depth of the SW. The superscripta
indicates that the simple relation m511(C21)/3 is used to fix m during
the optimisation, while b indicates that the range l is fixed.
T c* 5kT c b/ a VDW and P c* 5 P c b 2 / a VDW are the SAFT-VR critical points in
reduced units.
Substance¬
m¬
CH4
a
C2 H6¬
C3 H8¬
C4 H10
C5 H12¬
C6 H14¬
C7 H16
C8 H18¬
l
1
1.024¬
0.958¬
1.3 a
1.477¬
1.240¬
1.6 a
1.840¬
1.634¬
2a
1.990¬
2.835¬
2.3 a
2.312¬
2.056¬
2.6 a
2.430¬
2.116¬
3a
2.687¬
2.621¬
3.3 a
2.930¬
3.050¬
1.444¬
1.5b
1.431¬
1.448¬
1.5b
1.418¬
1.452¬
1.5b
1.441¬
1.501¬
1.5b
1.791¬
1.505¬
1.5b
1.443¬
1.552¬
1.5b
1.432¬
1.563¬
1.5b
1.486¬
1.574¬
1.5b
1.524¬
s ~Å!
e /k ~K!
T c*¬
P c*
3.670¬
3.510¬
3.736¬
3.788¬
3.617¬
3.908¬
3.873¬
3.706¬
3.910¬
3.887¬
3.890¬
3.282¬
3.931¬
3.944¬
4.161¬
3.920¬
4.092¬
4.361¬
3.933¬
4.137¬
4.188¬
3.945¬
4.188¬
4.105¬
168.8¬
146.4¬
349.1¬
241.8¬
209.5¬
264.2¬
261.9¬
229.0¬
269.8¬
257.2¬
256.9¬
132.3¬
265.0¬
269.2¬
316.0¬
250.4¬
288.2¬
349.1¬
251.3¬
297.0¬
308.5¬
250.3¬
304.2¬
285.3¬
0.150386¬
0.141336¬
0.150694¬
0.166737¬
0.164098¬
0.168570¬
0.179844¬
0.178288¬
0.180629¬
0.183526¬
0.183349¬
0.179181¬
0.192849¬
0.192950¬
0.194410¬
0.195698¬
0.196099¬
0.198074¬
0.202075¬
0.202382¬
0.202698¬
0.207646¬
0.207696¬
0.207314¬
0.00926859
0.00782506
0.00991756
0.00765300
0.00639648
0.00870246
0.00653363
0.00558603
0.00678226
0.00527521
0.00530230
0.00281339
0.00470330
0.00476882
0.00570210
0.00399982
0.00459515
0.00567886
0.00360600
0.00425083
0.00441532
0.00327096
0.00396231
0.00372699
with that for the arbitrary system,
a M 5a HS1 b
1b2
E ] f]
`
1
~ x ! SW
a 1 ~ x ! dx
x
E ] f]
~ x ! 2 SW
a 2 ~ x ! dx,¬
x
`
1
m¬
CF4
1a
1.056¬
1.37a
1.343¬
1.740a
1.697¬
C2 F6
C3 F8
l
1.287¬
1.291¬
1.339¬
1.332¬
1.359¬
1.352¬
it is clear that the high-temperature expansion will converge
faster for the arbitrary system with an attractive potential that
decreases with distance. In general the fluctuation term a 2
will make a smaller contribution to the free energy than in
the case of the square-well fluid. This is borne out in the case
of the Yukawa system where the first-order term a 1 already
provides a good description of the thermodynamic properties
and the effect of the fluctuation term a 2 is small ~see Section
IV A 3!.
~123!
VII. APPLICATION TO n-ALKANES AND
n-PERFLUOROALKANES
TABLE II. Optimised SW intermolecular potential parameters for
n-perfluoroalkanes obtained by a fit to experimental vapour-pressure and
saturated liquid density data from the triple point to the critical point ~Ref.
96!. m is the number of spherical segments in the model, l the SW range
parameter, s the diameter of each segment, and e is the well depth of the
SW.¬ The¬ superscript a indicates¬ that¬ the¬ simple¬ relation
m511(C21)0.37¬ is¬ used¬ to¬ fix¬ m¬ during¬ the¬ optimization.
T c* 5kT c b/ a VDW and P c* 5 P c b 2 / a VDW are the SAFT-VR critical points in
reduced units.
Substance¬
FIG. 7. Vapour pressures for methane compared with the SAFT-HS and
SAFT-VR predictions. The circles represent the experimental data ~Ref. 96!,
the dashed curves correspond to the SAFT-HS approach, and the solid
curves to the SAFT-VR approach for square-well segments. The parameters
are obtained by a fit to both experimental vapour pressures and saturated
liquid densities with fixed m51 ~see Table I!.
s ~Å!
e /k ~K!
T c*¬
P c*
4.346¬
4.260¬
4.436¬
4.476¬
4.474¬
4.524¬
278.6¬
269.6¬
289.0¬
295.8¬
298.8¬
306.1¬
0.197736¬
0.198721¬
0.194946¬
0.196070¬
0.202586¬
0.203069¬
0.0177391
0.0164518
0.0103974
0.0108675
0.00788251
0.00822553
In order to examine the predictive capabilities of the
SAFT-VR approach for real fluids we start by applying the
theory¬ to¬ the¬ phase¬ equilibria¬ of¬ n-alkanes¬ and
n-perfluoroalkanes. This is intended as an example; the
SAFT-VR equation of state is currently being used to determine the phase behaviour of a number of pure fluids and
mixtures involving associating chain molecules.
The SAFT-VR second-order expressions developed for
chains formed from square-well segments ~see Section
IV A 1! are used in this demonstration although the Sutherland or Yukawa potentials could also have been used. The
parameters s , e , l, and m of the square-well chain model are
optimised by fitting the calculated vapour-pressure curve and
saturated liquid densities to the experimental data for the
n-alkanes and n-perfluoroalkanes, from the triple to the critical point,96 using a simplex method.83 These fitted param-
J. Chem. Phys., Vol. 106, No. 10, 8 March 1997
Copyright ©2001. All Rights Reserved.
Gil-Villegas et al.: Statistical associating fluid theory
FIG. 8. Vapour-liquid coexistence curves for methane compared with the
SAFT-HS and SAFT-VR predictions. See Fig. 7 for details.
eters are reported in Tables I and II for the n-alkanes and
n-perfluoroalkanes, respectively. The parameters show the
rough tendency to increase with increasing number of carbon
atoms C, but whilst the range l appears to continue increasing, the size s and energy e of the segment-segment interaction appear to tend to a limiting value for the longer
chains. As expected, the diameters and the well-depth energy
of the segments are larger for the n-perfluoroalkanes than for
the n-alkanes, but the range l is smaller for the former molecules. It is important to note that the segments of our chain
molecules are united atom models so that the number of
segments in the chain does not represent the number of carbon atoms. Instead, the parameter m provides an indication
of the non-sphericity ~aspect ratio! of the molecule. By noting that for n-alkanes the carbon-carbon bond length is about
1
3 of the diameter of the methane molecule, a simple empirical¬ relationship¬ was¬ obtained¬ between¬ m¬ and¬ C:
m511(C21)/3.33,34 Within the SAFT-HS approach, this
simple relationship provides a good description of the critical
pressure and temperature for the n-alkanes although it does
not describe the anomalous progression of the critical pressure from methane to ethane.33,34 It is gratifying to see that a
free fit of m with the SAFT-VR approach to the experimental
vapour pressures and saturated liquid densities of the
n-alkanes leads to values which are consistent with the relationship between m and C, especially for the fixed l optimisations ~see Table I!. The value of the parameters obtained
for methane (m51) with the SAFT-VR approach are also in
close agreement with the parameters obtained by Chang and
Sandler68 with their second order perturbation theory incorporating the Percus-Yevick hard-sphere structure. The empirical relationship obtained earlier with the SAFT-HS ap-
4183
FIG. 9. Vapour pressures for propane (C53), butane (C54), and pentane
(C55) compared with the SAFT-HS and SAFT-VR predictions. The
circles represent the experimental data ~Ref. 96!, the dashed curves correspond to the SAFT-HS approach, and the solid curves to the SAFT-VR
approach for square-well segments. The parameters are obtained by a fit to
both experimental vapour pressures and saturated liquid densities with m
fixed as m511(C21)/3 ~see Table I!.
FIG. 10. Vapour-liquid coexistence curves for propane (C53), butane
(C54), and pentane (C55) compared with the SAFT-HS and SAFT-VR
predictions. See Fig. 9 for details.
J. Chem. Phys., Vol. 106, No. 10, 8 March 1997
Copyright ©2001. All Rights Reserved.
4184¬
Gil-Villegas et al.: Statistical associating fluid theory
FIG. 11. Vapour pressure curves for n-alkanes (C51 to 8! compared with
the SAFT-VR predictions. The experimental data are shown as circles ~Ref.
96!, and the solid curves correspond to the SAFT-VR approach for squarewell segments. The parameters are obtained by a fit to both experimental
vapour pressures and saturated liquid densities with m fixed as
m511(C21)/3 ~see Table I!.
proach for the n-perfluoroalkanes is m511(C21)0.37,34
which is found to be adequate from the fit of m to the experimental data using the SAFT-VR approach ~see Table II!.
Although the description obtained with the SAFT-HS approach is not as good as that with the SAFT-VR approach,
one advantage of the former is that the segment-segment
parameters can be used in a transferable way for all of the
homologous series:33–35 the parameters s 53.855 Å and
e mf/k5 a /b53135 K ~where this is now an integrated meanfield energy!, obtained for n-butane (m52) by fitting the
SAFT-HS values to the experimental vapour pressure and
saturated liquid density, can be used for other members in
the¬ series,¬ although¬ values¬ of s 53.650 Å and
e mf/k5 a /b52110 K from a separate fit have to be used for
methane (m51).
The vapour-pressure and coexistence density curves obtained for methane with the SAFT-HS and SAFT-VR approaches with fixed m51 are compared to the experimental
data96 in Figs. 7 and 8. The use of the SAFT-VR approach
with a variable range l greatly improves the accuracy of the
prediction due to the incorporation of the non-conformal effects of the range into the theory. As with any van der Waals
equation of state the SAFT approach is inadequate close to
the critical point: the critical temperature and especially the
critical pressure are overestimated. We compare the predictions of SAFT-HS and SAFT-VR using m511(C21)/3
with the experimental data96 for the vapour-liquid phase
equilibria of n-propane, n-butane and n-pentane in Figs. 9
and 10, and again the SAFT-VR approach is found to provide the better description for these chain molecules. Note
FIG. 12. Vapour-liquid coexistence curves for n-alkanes (C51 to 8! compared with theoretical predictions. See Fig. 11 for details.
that the transferable parameters obtained for n-butane are
used for all members in the SAFT-HS approach. The success
of the SAFT-VR approach in representing the phase behaviour of the homologous series of the n-alkanes from methane
FIG. 13. Vapour pressure curves for n-perfluoroalkanes (C51 to 3! compared with the SAFT-VR predictions. The experimental data are shown as
circles ~Ref. 96!, and the solid curves correspond to the SAFT-VR approach
for square-well segments. The parameters are obtained by a fit to both experimental vapour pressures and saturated liquid densities with m fixed as
m511(C21)0.37 ~see Table II!.
J. Chem. Phys., Vol. 106, No. 10, 8 March 1997
Copyright ©2001. All Rights Reserved.
Gil-Villegas et al.: Statistical associating fluid theory
4185
adequacy of the SAFT-VR approach in describing the phase
equilibria of chain molecules such as the n-alkanes and
n-perfluoroalkanes has been demonstrated. In future work we
will use the SAFT-VR equation of state developed here to
predict the phase equilibria of a wide range of systems ~pure
fluids and mixtures! containing associating chain molecules.
As for other augmented van der Waals equations of state, the
description of the critical region by the SAFT-VR approach
is inadequate. However, the correct behaviour at the critical
point can be incorporated into equations of state of this type
without changing the basic form of the expressions.97
ACKNOWLEDGMENTS
A.G.V. thanks the Engineering and Physical Sciences
Research Council ~EPSRC! for a Research Fellowship, A.G.
thanks Sheffield University and ICI Chemicals and Polymers
for the award of a Roberts-Boucher Scholarship, and P.J.W.
thanks the ICI Strategic Research Fund for funding a Senior
Research Fellowship. We also acknowledge support from the
European Commission ~CI1*-CT94-0132!, the Royal Society, and the Computational and ROPA Initiatives of the
EPSRC for the provision of computer hardware.
FIG. 14. Vapour-liquid coexistence curves for n-perfluoroalkanes (C51 to
3! compared with theoretical predictions. See Fig. 13 for details.
1
to octane is confirmed in Figs. 11 and 12. The trends for the
critical parameters are predicted by the theory ~including the
anomalous increase in the critical pressure in going from
methane to ethane!, but one should note that the parameters
were not fitted to the critical region and that the equation of
state is expected to be a less good representation there. A
similar performance is found with the SAFT-VR approach
for the n-perfluoroalkanes with m511(C21)0.37, as can
be seen from Figs. 13 and 14.
VIII. CONCLUSIONS
We have presented a version of the SAFT approach for
chain molecules formed from spherical segments with attractive potentials of variable range ~SAFT-VR!. The theory is
based on a general treatment of the dispersion forces using a
compact expression for the mean-attractive energy a 1 ~firstorder term! within a high-temperature perturbation expansion. Standard perturbation theory can be used to describe
the properties of the monomeric segments, including the contact value of the cavity function which is used to evaluate the
contribution to the free energy due to chain formation and
association. We have presented simple analytical expressions
for the Helmholtz free energy of chain molecules formed
from square-well, Sutherland, and Yukawa segments with
variable range. The Yukawa and Sutherland potentials are
important in that they can be used to model multipolar ~Sutherland! or electrolyte ~Yukawa! systems. Another advantage
of our SAFT-VR approach is that it can easily be extended to
mixtures. The corresponding expressions are compact and
essentially go beyond the van der Waals one-fluid level,82
which is commonly used for these types of theories. The
J. D. van der Waals, Thesis, University of Leiden, 1873; English translation and introductory essay by J. S. Rowlinson, 1988, On the Continuity of
Gaseous and Liquis States, Studies in Statistical Mechanics, Vol. 14
~North Holland, Amsterdam!.
2
L. L. Lee, Molecular Thermodynamics of Nonideal Fluids ~Butterworth,
Boston, 1988!.
3
N. F. Carnahan and K. E. Starling, J. Chem. Phys. 51, 635 ~1969!.
4
B. J. Alder and C. E. Hetch, J. Chem. Phys. 50, 2032 ~1969!.
5
M. L. McGlashan, Pure Appl. Chem. 57, 89 ~1985!; J. Chem. Thermodyn.
17, 301 ~1985!.
6
J. A. Barker and D. Henderson, J. Chem. Phys. 47, 2856 ~1967!.
7
J. A. Barker and D. Henderson, J. Chem. Phys. 47, 4714 ~1967!.
8
G. A. Mansoori and F. B. Canfield, J. Chem. Phys. 51, 4958 ~1969!.
9
J. D. Weeks, D. Chandler, and H. C. Andersen, J. Chem. Phys. 54, 5237
~1971!.
10
H. C. Andersen, D. Chandler, and J. D. Weeks, J. Chem. Phys. 56, 3812
~1972!.
11
M. S. Wertheim, J. Stat. Phys. 35, 19 ~1984!.
12
M. S. Wertheim, J. Stat. Phys. 35, 35 ~1984!.
13
M. S. Wertheim, J. Stat. Phys. 42, 459 ~1986!.
14
M. S. Wertheim, J. Stat. Phys. 42, 477 ~1986!.
15
M. S. Wertheim, J. Chem. Phys. 85, 2929 ~1986!.
16
M. S. Wertheim, J. Chem. Phys. 87, 7323 ~1987!.
17
W. G. Chapman, K. E. Gubbins, G. Jackson, and M. Radosz, Fluid Phase
Equilib. 52, 31 ~1989!.
18
W. G. Chapman, K. E. Gubbins, G. Jackson, and M. Radosz, Ind. Eng.
Chem. Res. 29, 1709 ~1990!.
19
S. H. Huang and M. Radosz, Ind. Eng. Chem. Res. 29, 2284 ~1990!.
20
S. H. Huang and M. Radosz, Ind. Eng. Chem. Res. 30, 1994 ~1991!.
21
M. L. Yu and Y. P. Chen, Fluid Phase Equilib. 94, 149 ~1994!.
22
S. J. Suresh and E. J. Beckman, Fluid Phase Equilib. 99, 219 ~1994!.
23
S. H. Huang and M. Radosz, Fluid Phase Equilib. 70, 33 ~1991!.
24
C. J. Gregg and M. Radosz, Fluid Phase Equilib. 86, 211 ~1993!.
25
I. G. Economou, C. J. Gregg, and M. Radosz, Ind. Eng. Chem. Res. 31,
2620 ~1992!.
26
C. J. Gregg, F. P. Stein, S. J. Chen, and M. Radosz, Ind. Eng. Chem. Res.
32, 1442 ~1993!.
27
B. M. Hasch, E. J. Maurer, L. F. Ansanelli, and M. A. McHugh, J. Chem.
Thermodyn. 26, 625 ~1994!.
28
S. W. Campbell, Fluid Phase Equilib. 102, 61 ~1994!.
29
P. J. Smits, I. G. Economou, C. J. Peters, and J. D. Arons, J. Phys. Chem.
98, 12080 ~1994!.
J. Chem. Phys., Vol. 106, No. 10, 8 March 1997
Copyright ©2001. All Rights Reserved.
4186¬
30
Gil-Villegas et al.: Statistical associating fluid theory
H. S. Byun, B. M. Hasch, and M. A. McHugh, Fluid Phase Equilib. 115,
179 ~1996!.
31
S.-J. Chen, I. G. Economou, and M. Radosz, Macromolecules 25, 4987
~1992!; C. J. Gregg, S. J. Chen, F. P. Stein, and M. Radosz, Fluid Phase
Equilib. 83, 375 ~1993!; C. J. Chen, I. G. Economou, and M. Radosz, ibid.
83, 391 ~1993!; C. K. Chen, M. A. Duran, and M. Radosz, Ind. Eng.
Chem. Res. 32, 3123 ~1993!; S. J. Suresh, R. M. Enik, and E. J. Beckman,
Macromolecules 27, 348 ~1994!; C. K. Chen, M. A. Duran, and M. Radosz, Ind .Eng. Chem. Res 33, 306 ~1994!; S. J. Chen, Y. C. Chiew, J. A.
Gardecki, S. Nilsen, and M. Radosz, J. Polymer Sci. B. 32, 1791 ~1994!;
D. Pradhan, C. K. Chen, and M. Radosz, Ind. Eng. Chem. Res. 33, 1984
~1994!; S. H. Lee, M. A. Lostracco, and M. A. McHugh, Macromolecules
27, 4652 ~1994!; C. J. Gregg, F. P. Stein, and M. Radosz, ibid., 4981 27,
4972 ~1994!; J. Phys. Chem. 98, 10634 ~1994!; C. S. Wu and Y. P. Chen,
Fluid Phase Equilib. 100, 103 ~1994!; B. M. Hasch and M. A. McHugh, J.
Polymer Sci. B 33, 715 ~1995!; Y. Xiong and E. Kiran, J. App. Polymer
Sci. 55, 1805 ~1995!; J. S. Chen, M. Banazsak, and M. Radosz, Macromolecules 28, 1812 ~1995!; B. M. Hasch, S. H. Lee, and M. A. McHugh,
J. Appl. Polymer Sci. 59, 1107 ~1996!; S. H. Lee, M. A. Lostracco, and M.
A. McHugh, Macromolecules 29, 1349 ~1996!; H. S. Byun, B. M. Hasch,
M. A. McHugh, F. O. Mahling, M. Busch, and M. Buback, Macromolecules 29, 1625 ~1996!.
32
W. G. Chapman, G. Jackson, and K. E. Gubbins, Mol. Phys. 65, 1057
~1988!.
33
G. Jackson and K. E. Gubbins, Pure Appl. Chem. 61, 1021 ~1989!.
34
A. L. Archer, M. D. Amos, G. Jackson, and I. A. McLure, Int. J. Thermophys. 17, 201 ~1996!.
35
A. Galindo, P. J. Whitehead, G. Jackson, and A. N. Burgess, J. Phys.
Chem. 100, 6781 ~1996!.
36
A. Galindo, P. J. Whitehead, G. Jackson, and A. N. Burgess, J. Phys.
Chem. ~in press!.
37
W. G. Chapman, J. Chem. Phys. 93, 4299 ~1990!.
38
D. Ghonasgi and W. G. Chapman, Mol. Phys. 80, 161 ~1993!.
39
J. K. Johnson, E. A. Muller, and K. E. Gubbins, J. Phys. Chem. 98, 6413
~1994!.
40
M. Banaszak, Y. C. Chiew, R. Olenick, and M. Radosz, J. Chem. Phys.
100, 3803 ~1994!.
41
M. Banaszak, Y. C. Chiew, and M. Radosz, Phys. Rev. E 48, 3760 ~1993!.
42
D. Ghonasgi and W. G. Chapman, J. Chem. Phys. 100, 6633 ~1994!.
43
F. W. Tavares, J. Chang, and S. I. Sandler, Mol. Phys. 86, 1451 ~1995!.
44
Y. H. Fu and S. I. Sandler, Ind. Eng. Chem. Res. 34, 1897 ~1995!.
45
D. G. Green and G. Jackson, J. Chem. Phys. 97, 8672 ~1992!.
46
R. Sear and G. Jackson, Mol. Phys. 82, 1033 ~1994!.
47
D. Ghonasgi, V. Perez, and W. G. Chapman, J. Chem. Phys. 101, 6880
~1994!.
48
D. Ghonasgi and W. G. Chapman, J. Chem. Phys. 102, 2585 ~1995!.
49
R. Sear and G. Jackson, Phys. Rev. E 50, 386 ~1994!.
50
R. Sear and G. Jackson, Mol. Phys. 87, 517 ~1996!.
51
R. Sear and G. Jackson, J. Chem. Phys. 105, 1113 ~1996!.
52
J. M. Prausnitz, R. N. Lichtentaler, and E. Gomes de Azevedo, Molecular
Thermodynamics of Fluid Phase Equilibria, 2nd ed. ~Prentice-Hall, New
Jersey, 1986!.
53
S. Beret and J. M. Prausnitz, AIChE J. 21, 1123 ~1975!.
54
C.-H. Kim, P. Vimalchand, M. D. Donohue, and S. I. Sandler, AIChE J.
32, 1726 ~1986!.
55
G. D. Ikonomou and M. D. Donohue, AIChE J. 32, 1716 ~1986!.
56
E. Waisman, Mol. Phys. 25, 45 ~1973!.
57
D. Henderson, E. Waisman, J. L. Lebowitz, and L. Blum, Mol. Phys. 35,
241 ~1978!.
P. T. Cummings and E. R. Smith, Chem. Phys. 42, 241 ~1979!.
P. T. Cummings and E. R. Smith, Mol. Phys. 38, 997 ~1979!.
60
P. T. Cummings, C. C. Wright, and E. R. Smith, Chem. Phys. Lett. 66,
278 ~1979!.
61
P. T. Cummings and G. Stell, J. Chem. Phys. 78, 1917 ~1983!.
62
F. Del Rı́o and L. Lira, Mol. Phys. 61, 275 ~1987!.
63
A. L. Benavides and F. del Rı́o, Mol. Phys. 68, 983 ~1989!.
64
A. Gil-Villegas and F. del Rı́o, Rev. Mex. Fis. 39, 526 ~1993!.
65
Y. P. Tang and B. C. Y. Lu, J. Chem. Phys. 99, 9828 ~1993!.
66
Y. P. Tang and B. C. Y. Lu, J. Chem. Phys. 100, 3079 ~1994!.
67
Y. P. Tang and B. C. Y. Lu, J. Chem. Phys. 100, 6665 ~1994!.
68
J. Chang and S. I. Sandler, Mol. Phys. 81, 745 ~1994!.
69
D. Henderson, L. Blum, J. P. Noworyta, J. Chem. Phys. 102, 4973 ~1995!.
70
A. Gil-Villegas, F. del Rı́o, and C. Vega, Phys. Rev. E 53, 2326 ~1996!.
71
G. Jackson, W. G. Chapman, and K. E. Gubbins, Mol. Phys. 65, 1 ~1988!.
72
J. A. Barker and D. Henderson, Rev. Mod. Phys. 48, 587 ~1975!.
73
R. W. Zwanzig, J. Chem. Phys. 22, 1420 ~1954!.
74
K. Rajagopal, D. W. Calvin, and T. M. Reed, J. Chem. Phys. 57, 4575
~1972!.
75
P. D. Fleming III and R. J. Brugman, AIChE J. 33, 729 ~1987!.
76
A. Gil-Villegas, Ph.D. thesis, Universidad Autónoma Metropolitana,
México, 1994.
77
F. Del Rı́o, A. L. Benavides, and Y. Guevara, Physica A 215, 10 ~1995!.
78
A. Gil-Villegas, F. del Rı́o, and A. L. Benavides, Fluid Phase Equilib.
119, 97 ~1996!.
79
R. Courant, Differential and Integral Calculus, Vol. 1 ~Blackie and Son,
London, 1957!.
80
T. Boublı́k, Mol. Phys. 59, 775 ~1986!.
81
A. Malijevsky and S. Labı́k, Mol. Phys. 60, 663 ~1987!.
82
J. S. Rowlinson and F. L. Swinton, Liquids and Liquid Mixtures, 3rd ed.
~Butterworth Scientific, London, 1982!.
83
W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery,
Numerical Recipes in Fortran, 1st ed. ~Cambridge University Press, London, 1986!.
84
L. Vega, E. de Miguel, L. F. Rull, G. Jackson, and I. A. McLure, J. Chem.
Phys. 96, 2296 ~1992!.
85
G. Stell, J. C. Raisaiah, and H. Narang, Mol. Phys. 23, 393 ~1972!.
86
E. Lomba and N. G. Almarza, J. Chem. Phys., 100, 8367 ~1994!.
87
T. M. Reed and K. E. Gubbins, Applied Statistical Mechanics ~McGrawHill, Tokyo, 1973!.
88
A. Z. Panagiotopoulos, Mol. Phys. 61, 813 ~1987!.
89
J. K. Johnson, J. A. Zollweg, and K. E. Gubbins, Mol. Phys. 78, 591
~1993!.
90
M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions
~Dover, New York, 1965!.
91
A. Yethiraj and C. K. Hall, Mol. Phys. 72, 619 ~1991!.
92
F. A. Escobedo and J. J. de Pablo, Mol. Phys. 87, 347 ~1996!.
93
P. J. Leonard, D. Henderson, and J. A. Barker, Trans. Faraday Soc. 66,
2439 ~1970!.
94
T. Boublı́k, J. Chem. Phys. 53, 471 ~1970!.
95
G. A. Mansoori, N. F. Carnahan, K. E. Starling, and T. W. Leland, J.
Chem. Phys. 54, 1523 ~1971!.
96
B. D. Smith and R. Srivastava, Thermodynamic Data for Pure Compounds
~Elsevier, Amsterdam, 1986!.
97
A. Van Pelt, G. X. Jin, and J. V. Sengers, Int. J. Thermophys. 15, 687
~1994!.
58
59
J. Chem. Phys., Vol. 106, No. 10, 8 March 1997
Copyright ©2001. All Rights Reserved.