MOLECULA R PHYSICS, 1998, VOL. 94, NO. 6, 937± 948
Fluids of pseudo-hard bodies
II. Reference models for water, methanol, and ammonia
By MILAN PRÏEDOTA 1 , IVO NEZBEDA 1,2 , and YURIJ V. KALYUZHNYI3
1
E. Ha la Laboratory of Thermodynamics, Institute of Chemistry Process
Fundamentals, Academy of Sciences, 165 02 Prague 6 ± Suchdol, Czech Republic
2
Department of Physics, J. E. PurkyneÏ University, 400 96 U
 stõ n. Lab.,
Czech Republic
3
Institute of Condensed Matter Physics, Ukrainian National Academy of Sciences,
290011 Lviv, Ukraine
( Received 5 February 1998; revised version accepted 24 March 1998 )
Pseudo-hard body ¯ uids resulting from extended primitive models of water, methanol, and
ammonia have been investigated both by computer simulations and theory for a number of
geometrical parameters. It is shown that none of the existing equations of state and integral
equations for the site± site correlation functions is able to describe the properties of the pseudohard body ¯ uids reasonably accurately. For the equation of state an accurate semi-empirical
method is proposed and for the site± site correlation functions the reference average Mayer
function perturbation theory has been found to perform at least qualitatively correctly, which
is not the case with Ornstein± Zernike equation based theories.
1.
Introduction
Recently we have introduced a new type of hard body
(`pseudo-hard’ bodies; PHBs) [1]; these arise from
extended models of associated ¯ uids [2] in connection
with the thermodynamic perturbation theory of
Wertheim [3]. The bodies may be characterized as
fused hard sphere models with sites of three kinds:
(i) real hard sphere sites forming a hard core of particles,
and (ii) embedded sites of two kinds with simple nonadditive site± site interactions: the sites interact either as
hard spheres (pairs of like sites) or do not interact at all
(pairs of unlike sites).
Despite its simplicity it is not trivial to predict properties of this model ¯ uid, which provides a real challenge
for theory. For instance, some typical properties of
common hard bodies, e.g., excluded co-volume, may
hardly be de® ned in terms of the individual bodies; similarly, the common de® nition of the packing fraction
loses its meaning for PHB ¯ uids.
In [1] we considered brie¯ y only one type of PHB
model, a descendant of a ® ve-site model of water, and
attempted to determine both its thermodynamic and
structure properties from existing theories. Whereas
for the former properties it seemed possible to devise a
method yielding reasonably accurate results, the results
for the structure were not very satisfactory. Owing to the
high symmetry of the model, the centre± centre pair correlation functions calculated from the RAM perturbation theory [4] are in reasonable agreement with
0026± 8976/98 $12 . 00
Ñ
simulation results but results for the correlation functions involving embedded sites are unsatisfactory. The
disagreement becomes even worse for asymmetric
models with unpaired sites.
The PHB model is undoubtedly of interest in its own
right, especially from the theoretical point of view, but
its main importance lies in the ® eld of associated ¯ uids.
It has become common to model, for both theoretical
and practical purposes, associated ¯ uids by primitive
models [5, 6] or, more accurately, by extended primitive
models [2, 7]. An accurate description of all the properties of the PHB ¯ uids makes it possible to implement
thermodynamic perturbation theory for extended
models [8], which in turn enables us to develop a perturbation theory of associated ¯ uids using the extended
model as a reference system [9]. On a simpler level, the
equation of state of these ¯ uids forms a core of analytical molecular-based equations of state for practical
applications [10]. For these reasons we do not consider
in this paper arbitrary models but only those originating
from models of typical associated ¯ uids: water,
methanol, and ammonia. We apply the known methods
and their modi® cations in order to develop an equation
of state in a closed anaytical form. The structural properties are determined using both a perturbation theory
[4] and Ornstein± Zernike equation based theories [11].
The variety of models considered makes it possible to
check the accuracy of di€ erent theoretical methods for a
wide range of model geometries and thermodynamic
1998 Taylor & Francis Ltd.
938
M. PrÏedota et al.
conditions and to draw certain general conclusions as
regards the applicability and accuracy of the methods
considered. It is possible to claim that, in general, none
of the existing methods performs satisfactorily for PHB
¯ uids. As regards the equation of state, it may happen
that a certain theoretical method yields good results for
one model but it then usually fails for other models.
Although usually it is possible to link the failure of
theory to the number of embedded sites, it is hard to
draw any de® nite conclusions. Concerning the correlation functions, RAM perturbation theory [4] seems to
provide the most consistent results, although its performance also varies from very good to rather poor.
The integral equations considered, the RISM [12] and
proper RISM [13, 14] theories, seem very unreliable,
providing in many cases even unphysical results.
2.
M ode ls, basic de® nitions, and computational details
A PHB ¯ uid is de® ned, in general, by a rigid fused
hard sphere core (made up of true hard sphere sites
referred to as `centres’ ) with embedded hard sphere sites
of two kinds and by the total pair potential of the form
u
(1, 2) = u core ( R1 , R2 , X
1
,X
2)
+
å{ , }
| -
u hs ( r 1,i
|
r 2,j ; s
ij
),
i j
( 1)
where u core denotes a hard body potential of the core, Ri
and X i denote the position vector of a reference point
within the hard core and its orientation, respectively, r k ,j
is the position vector of site j on molecule k , the summation runs over all pairs of the like embedded sites only,
and the hard sphere interaction, u hs , is given
u hs (r 12
; s ) = +¥
for
r 12
<s
=0
for
r 12
>s .
,
( 2)
Throughout the paper all (embedded) site± site interactions are considered identical, i.e., s i j º s R . In order
to maintain a link with primitive models of associated
¯ uids, we will refer to the embedded sites as + and sites. For completeness, the structure of the monomer
(i.e., the core and arrangement of sites) must be de® ned.
In this paper we will consider several models, all descending from realistic potential models of water,
methanol, and ammonia; the hard core of all these
models is either a hard sphere of diameter s c º s O =
s N = 1, or a homonuclear dumbbell made up of two
spheres of diameter s O = s C = 1 with the site± site
span l * º l /s O = 0. 45 (see ® gure 1).
Three- and ® ve-site models of water, denoted as W3
and W5, respectively, have their origin in the TIP3 [15]
and ST2 [16] potentials. As regards methanol, at the
level of primitive models only a ® ve-site monomer, descending from the TIP-model [19] has been considered
Figure 1. PHB models descending from realistic pair potentials: W3 and W5 models descending from the TIP3 and
ST2 potentials, respectively; ® ve-site model Amm5 corresponding to a model of ammonia; models MeOH3 and
MeOH4 descending from di€ erent site± site models of
methanol. All o€ -centre sites are arranged tetrahedrally
on the surface of a hard sphere of unit diameter.
so far in simulations [5, 17]. However, in applications to
real mixtures a four-site model (MeOH4), obtained from
the ® ve-site model by replacing the dumbbell core by a
hard sphere, has been considered as well for its simplicity [18]. At the level of realistic potentials, other monomers also have been considered. For instance, Jorgensen
[15] considers only a three-site model with charges on
the carbon and oxygen atoms, and on the hydrogen of
the hydroxyl group. Since the charges on the C and O
atoms are of the same sign, the corresponding PHB
model (MeOH3) consists of a hard dumbbell core and
only one embedded o€ -centre site. We are aware of only
one rigid monomer considered for ammonia [20]. It is a
® ve-site model mimicking the position of four atoms and
a dummy site in the position of a lone electron pair. At
the level of primitive models, this model has been considered also as a ® ve-site model with tetrahedrally
arranged embedded sites, see ® gure 1.
All the PHB models shown in ® gure 1 will be considered in this paper. In order to obtain as much `experimental’ data as possible and to trace certain trends as
parameters of the models vary, we use the models not only
with parameters corresponding to realistic parent models
(largest values of s R), but also with a smaller value of s R
for which the e€ ect of non-additive site± site interaction
may be considered as a small perturbation only. A summary of all the models considered is given in table 1.
2.1. Thermodynamic properties
For hard body ¯ uids very important quantities are
virial coe cients, B i , often used to derive an equation
939
Fluids of pseudo-hard bodies. II
Table 1. PHB models depicted in ® gure 1 and considered in
this paper along with their virial coe cients, B *i =
B i /6 core , where 6 core is the volume of the hard core,
and a parameter of non-sphericity, a , equation (4);
s R º s ++ = s - - .
B*
2
B*
B*
z ref
a
Model
s
W3± 4
W3-6
W3-8
0.4
0.6
0.8
4.171
4.890
6.683
10.74
14.26
26.20
19.87
28.28
67.80
1.057
1.297
1.894
W5-4
W5-6
W5.8
0.4
0.6
0.8
4.339
5.768
9.130
11.48
18.91
46.36
21.48
37.66
132.8
1.113
1.589
2.710
Amm5-4
Amm5-6
Amm5-8
0.4
0.6
0.8
4.424
6.185
9.966
11.90
22.05
57.14
22.43
49.80
203.7
1.141
1.728
2.989
MeOH3-4
MeOH3-6
MeOH3-8
0.4
0.6
0.8
4.293
4.391
4.659
11.28
11.70
12.93
21.17
21.98
24.46
1.097
1.130
1.219
MeOH4-4
MeOH4-6
MeOH4-8
0.4
0.6
0.8
4.212
5.111
7.366
10.91
15.29
30.66
20.26
29.89
76.91
1.071
1.370
2.122
R
3
where subscript `ref ’ denotes the properties of a reference ¯ uid. The typical choice for the reference ¯ uid is
that of hard spheres, i.e.,
4
b
P
/q
º
z
=
1 + (3a
- 2) h
+ (3a 2 - 3a + 1) h
(1 - h ) 3
2
- a 2h 3 ,
( 3)
where P is the pressure, q is the number-density,
b = 1 /k B T , T is the temperature, k B is the Boltzmann
constant, and h = q 6 PHB is the packing fraction with
6 PHB the volume of the pseudo-hard body. The parameter a may be obtained using the properties of an
assigned hard convex body or, in a simpler way, via
the second virial coe cient [21],
6
B2/
PHB
= 1 + 3a
( 4)
Another possibility to make use of the knowledge of
virial coe cients is via the truncated perturbed virial
expansion,
z PVE (q
) = z ref (q
ref ) +
å
M
i =2
(B i q i - 1 -
i- 1
B ref ,i q ref )
,
( 5)
z hs
1+ h + h (1 - h ) 3
h
2
=
3
,
(6)
where h = q 6 hs , and 6 hs , is the volume of the reference
hard spheres, 6 hs = p6 s 3hs . If an ideal gas is chosen for
the reference, then equation (5) is identical to the
common virial expansion. The common virial expansion
may also be summed entirely if a certain form for the
virial coe cients is assumed. Actually, the origin of
many hard body EOSs (including equation (3)) may be
traced back to the following general form of the virial
coe cients [21]:
Bi
= 1 + f 2 (a ) (i - 1) + f 3 (a ) ( i - 1) (i - 2)
+ f 4 (a ) (i - 2) (i - 3) ,
for i >
z VE
of state (EOS) in an analytical form [21]. These coe cients are computed easily using the standard numerical
i 1
Monte Carlo method formulated for ratios B i /B 2- ,
i>
3 [21]. The number and complexity of the graphs
contributing to B i increase with increasing i , and therefore the coe cients most frequently computed directly
are only those for i < 4.
The most straightforward method for deriving an
EOS of a hard body ¯ uid is to use a parameter of
non-sphericity a in combination with the improved
scaled-particle theory EOS (ISPT) [21]:
º
(7)
2, which implies that
1
=
1-
h
+ f2
h
(1 - h )
2h
f
2+ 3
2
(1 - h )
f
3+ 4
2h
3
(1 - h ) 3
.
(8)
If the ® rst four virial coe cients are known then the
parameters f i may be determined from the relations following from equation (7):
f2
= B 2 - 1,
f3
= (B 3 + 1 - 2B 2 ) /2,
f4
= (B 4 - 3B 3 + 3B 2 - 1) /2.
(9)
2.2. Structural properties
The structure is described conveniently by a set of
site± site correlation functions,
g i j (r i j )
=
ò
r ij =const
, 2) d (1) d (2) ,
g (1
( 10)
where g (1, 2) is the full pair correlation function. At low
densities g i j approaches the average Boltzmann factor
(ABF) in the corresponding site± site frame [22],
g i j (r i j )
®
e i j (r i j )
=
ò
r ij =const
exp
[- b
u
]
(1, 2) d (1) d (2) .
( 11)
Integration of any e i j (r i j ) over separations yields the
second virial coe cient,
B2
= - 2p
ò [k
e i j (r i j )
l-
]
1 r 2i j dr i j .
( 12)
940
M. PrÏedota et al.
From the point of view of theory, the most important
g i j (r i j ) is the centre± centre correlation function because
of its use in perturbation theories.
The average Boltzmann factor is also the key quantity
in the RAM perturbation theory of molecular ¯ uids [4].
It de® nes a simple ¯ uid reference by
-b
= ln e i j (r ) ,
u RAM (r )
(13)
the properties of which are used to approximate the
structure of the original molecular ¯ uid. The full pair
correlation function is approximated by
- b u (1,2) y
g (1, 2) = e
(r ) ,
(14)
RAM
from which it follows that
-b
g (r ) = e
ij
^
^
u ij (r )
y RAM (r )
,
(15)
^
^
^
^
= S (k ) c^ (k ) S T (k ) + S (k ) c^ (k ) q h (k ) ,
c^i j (k )
^
S i j (k )
=
=
=
æ
è
(
(
^
h i 0 j 0 (k )
h i 0 j 1 (k )
^
^
(16)
h i 1 j 1 (k )
c^i 0 j 0 (k )
c^i 0 j 1 (k )
c^
c^
d
^
(k )
i 1j 1
0
ij
S i j (k )
ö,
^
h i 1 j 0 (k )
i1j0
d
ij
)
= f i j (r )
[
]
t ij (r )
( 17)
+1
and
where q = d i j q is the diagonal matrix with the number
^
density q being the diagonal element, and h (k ) , c^ (k ) , and
^
S (k ) are matrices whose elements are, in the case of the
RISM equation, the Fourier transforms of the total,
direct and intramolecular site± site correlation functions,
^
^
h i j (k ) , c^i j (k ) and S i j (k ) , respectively, and in the case of
the pRISM theory matrices [14, 23]
h i j (k )
c i j (r )
12
In the above equations y RAM is the background correlation function of the spherical RAM-reference ¯ uid.
Another theory which may potentially provide the
site± site correlation functions, and hence also the
thermodynamic properties of the PHB ¯ uids, is the
site± site integral equation theory [11]. In this paper we
consider two versions of this theory: (i) the original
reference interaction site model ( RISM); [12]; and (ii)
the `proper’ RISM equation (pRISM) [13, 14] closed
by the corresponding versions of the Percus± Yevick
type (PY) and hypernetted chain type (HNC) closures.
The set of the Ornstein± Zernike (OZ)-like integral
equations in both cases can be written in the form
h (k )
[14, 23]. Indices i and j denote the species of the sites,
and their subindices a and b denote the bonded states of
the corresponding sites. Value a = 0 denotes the
unbonded (or atomic-like) state, and a = 1 denotes the
bonded state. This notation is consistent also with the
notation introduced in the two-density theory for association ¯ uids [3], of which the present pRISM theory is
only a special case for the case of complete association
[24].
The PY and HNC closures read as, respectively,
(k )
,
^
U
ij
-
(r ) + t ij (r )
t i j (r )
-
1
( 18)
]
( 19)
in the case of the RISM theory, and
c ia
jb
(r ) = f i j (r )
[
tia
jb
(r ) + d
a 0d b 0
and
c ia
jb
(r ) = e- b
u i j (r ) + t u
+ d a 1d
0j 0
(r )
{d a
b 0 t t 1 j 0 (r )
+ t i 0 j 1 (r ) t i 1 j 0 (r )
0d b 0
+ d a 0d
[
b 1 t i 1 j 1 (r )
+ d a 1d
]} -
b 1 t u 0 j 1 (r )
t ia
jb
(r ) - d
a 0d b 0
( 20)
in the case of the pRISM equation. Here d a b is the
Kronecker delta, t i j (r ) = h i j (r ) - c i j (r ) , t i a j b (r ) = h i a j b (r ) c i a j b (r ) , f i j = exp - b u i j (r ) - 1, and the overall site± site
correlation function h i j (r ) is related to the partial site±
site correlation functions by
[
]
h i j (r )
=
å
1
a ,b =0
h ia
jb
(r ) .
( 21)
Once the site± site pair distribution functions g i j (r ) =
+ 1 are known, it is possible to determine the thermodynamic properties of the system, speci® cally the
equation of state, using the compressibility equation
h i j (r )
ø
)
= e- b
c i j (r )
q
,
T
b
[q ]
ij
=
( )
q
q
q
0
,
involving the partial total h i a j b ( k ) , direct c^i a j b (k ) and
^
^
intramolecular S i a j b (k ) correlation functions. S i j (k ) =
(sin k L i j ) /k L i j , L i j is the intramolecular distance between
the sites of species i and j and we are using a slightly
modi® ed version of the notation of Rossky and Chiles
x =1+ q
^
h i j (k
= 0) ,
( 22)
where x T is the isothermal compressibility and i and j
refer to any pair of sites.
Under the HNC closure conditions, the pressure can
be calculated by the direct integration of relation (22)
over density, while using the PY approximation one can
avoid such a procedure and utilize the corresponding
extension of the Baxter’ s expression for the PY compressibility pressure [25]. Such an expression is available in
941
Fluids of pseudo-hard bodies. II
the case of the RISM-PY approximation [26]
z
= 1 + 12 q
å
ò[
2
c ij (r )
ij
( ) ò{
1
+
2p
¿ (r )
Tr
[
¿ ( ) = 1 - exp [b
[
]
^
q S (k ) c^ (k )
+ g CC (s
+ g ++ (s
+ ln det 1 - q S (k ) c^ (k ) } dk ,
(23)
]
u i j (r ) , while in the case of
where i j r
pRISM-PY theory it could be derived using the equivalence between the pRISM equation and two-density
Wertheim’s OZ equation for associated ¯ uids [3, 24].
Unfortunately, the value of x T obtained using pRISMPY or pRISM-HNC theories via relation (22) for molecules with a number of sites larger than two depends
strongly on the choice of the pair of sites. For this
reasons pressure calculations are carried out here using
only the RISM-PY theory.
2.3. Computational details
To perform computer simulations on PHB models, we
used the standard Metropolis Monte Carlo (MC)
method in an N V T ensemble with periodic boundary
conditions [27]. A typical number of particles in the
central cell was N = 216, although several simulations
were carried out also with N = 512 to check the accuracy of the results. We have also computed the structural and orientation-order parameters in order to check
the spatial and orientational arrangement of particles,
and to keep the development of the simulated ¯ uids
under control [28].
The general formula for the compressibility factor of
the PHB ¯ uid with a spherical core in an N V T ensemble
takes on the form [29]
z
=1+
2p q
3
{g
cc
(s c ) s
3
2
R) s R
+ N - - g - - (s
2
R
R) s
k
k
Rc c m ++
Rc c m - -
ls
R
l s },
R
{gOO (s
(24)
where g ij (s ) is the value of the i - j intermolecular correlation function extrapolated to contact, N i j is the
number of i - j pairs, Rc c is the vector between the reference points of molecules, t i j is the unit vector in the
direction of i - j sites, and k . l s denotes the ensemble
average extrapolated to contact. The reference site, c ,
is that of nitrogen in the case of Amm5 model, and
the oxygen site in other cases. For the MeOH3 model
with the dumbbell core the corresponding expression
assumes the form
3
O) s O
2
O) s O
k
O) s
2
O
R) s
2
R
k
ROOm
k
ROO m
l
CO s
l
CC s
ROO m ++
O
O
l s },
R
( 25)
where subscripts C and O refer to the carbon and
oxygen sites, respectively. In order to reduce the
number of extrapolations that have to be performed,
the entire expression on the RHS of equation (24),
arising from the repulsion interactions between the
embedded sites, was computed in the course of the simulation. It is a smooth and nearly linear function near the
contact separation, so that its extrapolation is very accurate. Consequently, when a comparison of values of z
obtained from two di€ erent simulations at the same
state point was carried out, the di€ erence was always
less than 1%. The number of relaxation cycles varied
5
5
from 10 to 2 ´ 10 depending on the model and h .
The number of accumulation cycles was one half of
the number of relaxation cycles.
The virial coe cients were obtained using the MC
integration [21]. Ten million con® gurations of three molecules giving nonzero contribution to B 4 were generated.
4
For ABFs, for each ® xed separation of given sites 10
con® gurations of two molecules were generated.
The background correlation function of the spherical
RAM reference ¯ uid can be accurately determined
numerically using the well known optimized RHNC
theory with an `exact’ hard sphere bridge function [30].
Finally, solution of the set of equations (16) together
with PY and HNC closure conditions (17)± (20) was
obtained numerically using a combination of the direct
iteration and Newton± Raphson methods due to Labik
et al. [31].
3.
c
+ N ++ g ++ (s
2p q
3
+ 2g CO (s
]
]
^
=1+
2c i j (r ) dr
ij
3
z
Results and discussion
We performed MC simulations on each model for a
number of densities. Since the properties of the models
vary continuously with s R, we considered in simulations
only models with two extreme values of s R, one with
s R = 0. 4 and the other with a large realistic value 0.8.
We also applied to these models all theories outlined in
the preceding section.
The main goal of our investigations has been a theoretical estimate of the properties of PHB ¯ uids: (i) equation of state in a closed analytical form, and (ii) the site±
site correlation functions with the emphasis on the
centre± centre correlation function for its further use in
a perturbation theory. To reach the ® rst goal we have
followed the routes outlined in section 2.1 and computed
the ® rst four virial coe cients not only for all the
942
M. PrÏedota et al.
models considered but also for models with an intermediate value s R = 0. 6. For an insight into the structure
and as the prerequisite for the RAM theory, we computed also the ABFs in all site± site frames of interest.
3.1. Thermodynamic properties
The virial coe cients, summarized in table 1, are proportional to appropriate powers of a parameter characterizing the size of the molecules. For hard body ¯ uids it
is convenient to use the volume of the body as such a
parameter, cf. equation (4). The only well de® ned
volume-like quantity for the PHB models seems to be
the volume of the hard core, 6 core , which has been used
therefore to reduce the virial coe cients and to determine the packing fraction h . This is also the reason why
some coe cients and, consequently, also a for some
models, seem unrealistically large.
Compressibility factors obtained by simulations are
summarized in table 2. The simulations were carried
out up to densities at which a homogeneous ¯ uid
phase was guaranteed. Blanks in the table for s R = 0. 8
mean that the system was beyond this limit. Regarding
MeOH3, for reasons given in detail below, we simulated
this model with s R = 0. 4 only for one density.
When applying theoretical methods up to the PHB
¯ uids we should distinguish two groups of models with
respect to estimated e€ ects of the embedded spheres.
When the diameter of the sphere is relatively small or
when there is only one embedded sphere (the case of
MeOH3), the impact of the embedded spheres may be
considered as a small perturbation only. On the other
hand, large embedded spheres may a€ ect the properties
of the PHB ¯ uids considerably. These intuitive considerations are veri® ed in ® gures 2 and 3 for selected
models. For the remaining models the results are
similar, with a slightly better agreement between
theory and experiment.
Application of equation (3) to PHB ¯ uids poses no
problems if a is determined from (4) and 6 PHB is
approximated by 6 core . As seen from ® gure 2, this
route gives for s R = 0. 4 results in agreement with the
simulation data within experimental errors. However,
for larger values of s R its accuracy deteriorates rapidly,
see ® gure 3, underestimating the data. An exception is
the MeOH3 model for which it slightly overestimates the
data for s R = 0. 8.
The truncated virial expansion (5) with M = 4 and the
hard core reference at the density of the PHB ¯ uid is
expected, due to the neglect of higher virial coe cients,
to underestimate the simulation results, and this has
been con® rmed. An exception is again the MeOH3
model for which this route provides, for reasons given
above, results in excellent agreement with the experimental data even for s R = 0. 8. This is also the reason
Table 2. Compressibility factors of the
PHB ¯ uids from computer simulations
as a function of h , h = q 6 core .
b
h
P /q
\s R
0.4
0.8
W3
0.20
0.25
0.30
0.35
0.40
2.47
3.20
4.17
5.48
7.30
4.20
6.06
8.80
12.61
W5
0.15
0.20
0.25
0.30
0.35
0.40
2.01
2.58
3.32
4.34
5.74
7.65
3.87
5.89
8.62
12.56
Amm5
0.10
0.15
0.20
0.25
0.30
0.35
2.03
2.61
3.40
4.47
5.89
MeOH3
0.20
0.25
0.30
0.35
0.40
0.45
Model
MeOH4
0.10
0.15
0.20
0.25
0.30
0.35
0.40
7.67
1.96
2.49
3.20
4.19
5.53
7.40
2.80
4.76
8.17
14.13
2.82
3.55
4.67
6.20
8.23
11.21
2.12
3.10
4.50
6.53
9.34
13.33
why we simulated this model with s R = 0. 4 for one
packing fraction only.
Finally, the resummed virial expansion (6) tends to
overestimate the compressibility factors signi® cantly,
and only in the case of Amm5 with s R = 0. 8 does it
slightly underestimate it.
Since for s R = 0. 8 the size of the embedded spheres is
comparable with the central ones, the e€ ective nonspherical shape of particles becomes signi® cant, and it
is thus tempting to try to use in this case for the reference a fused hard sphere model obtained from the PHB
model by letting all embedded sites become true hard
sphere sites. Implementation of this route requires the
compressibility factor of the fused hard sphere ¯ uids
and their virial coe cients, which can be easily and
quite accurately obtained from the Boublik± Nezbeda
Fluids of pseudo-hard bodies. II
Figure 2. Comparison of the compressibility factors of the
PHB models obtained from computer simulation ( · ) and
di€ erent analytical equations of state for Amm5 with
s R = 0. 4 and MeOH3 with s R = 0. 8: ´´´´´´, ISPT; Ð Ð ,
perturbed virial expansion (5) with M = 4 and a hard
sphere reference; and Ð Ð , resummed virial expansion
(8).
Figure 3.
As for ® gure 2 for s
R
943
method [21]. However, it turns out that this route
strongly overestimates the e€ ect of the repulsive interactions and the results are practically useless. For this
reason the compressibility factors obtained via this route
are not shown in the ® gures.
Two possible ways to improve the performance of the
above discussed analytical equations of state for large
s R are (i) to include higher virial coe cients, and (ii) to
use a more appropriate reference or at least to estimate
more accurately the volume of pseudo-hard particles.
Since the ® rst possibility has only a very limited applicability (to compute virial coe cients beyond the ® fth
one for a number of models is practically out of the
question), we have attempted to follow the second way
by accounting for the number and type of embedded
sites by an empirical modi® cation of the volume of the
hard core.
It is clear that the embedded spheres exclude a certain
volume and thus that the above reference hard core
¯ uids have been considered at too low densities. Since
for the obvious reasons the actual volume of molecules
is not known, we may account for this fact by considering q hs in the form q hs = F q PHB , where (empirical ) parameter F provides an estimate of a certain `e€ ective’
volume of the PHB molecule, 6 PHB < F 6 hs . In order
to get ® rst a feeling for values of F , we have ® tted it so as
to reproduce the data as faithfully as possible. We have
obtained F W3 = 1. 28, F W5 = 1. 24, F MeOH4 = 1. 27, and
F Amm5 = 1. 90. As seen from ® gure 3, the perturbed
= 0. 8: - ´´- ´´- , perturbed virial expansion (5) with M = 4 and an e€ ective hard sphere reference.
944
M. PrÏedota et al.
expansion (5) with these F ’s reproduces the data excellently. The question is, whether the F ’s can be given a
physical meaning with a possibility of their a priori prediction.
If the embedded sites were only `ghost’ sites without
any site± site interaction, then 6 PHB would equal the
volume of the hard core, u 6 c . On the other hand, if
all the embedded sites were of the same kind, i.e., they
were true hard sphere sites, then 6 PHB would equal the
volume of a fused hard sphere particle, 6 FHS ,
6 FHS = (1 + ¢) 6 c . Let us try now to estimate F by
interpolating between these two extreme cases by considering the Amm5 model. This model has four
embedded sites and in the limiting fused hard sphere
model there are therefore 16 site± site repulsive interactions. Since in the PHB model only three sites are of
2
the same kind, we have only 3 + 1 = 10 additional
Figure 4. The
average
Boltzmann factors of the
PHB models W5 (left column) and Amm5 (right
column) for s R = 0. 4
(Ð Ð ) and (- - -), and
s R = 0. 8 ( ´´´ ´´´) and
( - ´´ - ´´ -) in di€ erent
site± site frames. Long
ticks on the abscissa
denote the locations of
cusps.
interactions. By modifying the fused hard sphere term
¢ by the fraction of the actual interactions, we get
6 PHB < (1 + 10
16 ¢ ) 6 c and hence (since ¢ = 1. 393 for
Amm5 model) F < 1. 87 which is very close to the
value of 1.90 obtained by ® tting. Unfortunately, by
applying the same procedure to other models, we get
F W3 = 1. 696, F W5 = 1. 696, and F MeOH4 = 1. 58. An
explanation for this failure is simple. The closer the
model is to a real fused hard sphere model, the better
the results obtained and this is the case for the Amm5
model for which the o€ -centre repulsions are predominant. In other models the lack of centre± embedded site
repulsions is substantial for the behaviour of the model,
and this fact must be incorporated into a better approximation for the volume of PHBs.
In principle, one can determine the compressibility
factor also from equation (23) using the results of the
Fluids of pseudo-hard bodies. II
945
Figure 5. Comparison of the
site± site correlation functions of the W3 model at
h = 0. 2 (left column) and
h = 0. 35 (right column)
obtained from computer
simulations ( · ) and di€ erent theories for s R = 0. 4:
. . . . . .,
Ð Ð ,
RAM;
RISM-PY; - - -, RISMHNC: - ´´ - ´´ - , CSLPY; and Ð
Ð , CSLHNC.
RISM-type theory for the site± site correlation functions.
We followed this route but, with very poor results for
g i j ’s (see below) , no results for z are shown or discussed.
3.2. Structural properties
From the density expansion it follows that the ABF in
appropriate reference frames provides the low density
limit of the site± site correlation function. In the case
of hard body ¯ uids, whose properties may be explained
to a large extent by purely geometrical considerations,
the site± site correlation function often follows the course
of the corresponding ABF even at higher densities. We
begin therefore the discussion of the structural properties of PHB ¯ uids by examining the ABFs in all site± site
frames.
The ABFs for di€ erent models are quite similar and
their typical shape is shown in ® gure 4 for the W5 and
Amm5 models for di€ erent s R. As seen, there is no
signi® cant qualitative di€ erence between ABFs for
s R = 0. 4 and s R = 0. 8. For the models with the spherical core the centre± centre ABF is always a smooth
function. All other ABFs are characterized by the presence of cusps (slope discontinuities). The origin and
location of the cusps for the models of water were discussed at length in [2] and therefore we refer the reader
to this paper for details.
Because of the relative simplicity of ABFs and their
potential use in a perturbation theory, it may be
tempting to express them analytically. This is relatively
simple for small values of s R when a simultaneous
overlap of only a pair of spheres must be considered.
946
M. PrÏedota et al.
Figure 6. As for ® gure 5 for
s R = 0. 8.
However, in the most interesting case of larger s R a
simultaneous overlap of more than two spheres occurs
and analytical calculations become prohibitively complex. The required ABFs were therefore determined
numerically.
The site± site correlation functions depend on both s R
and density. Because of the number of models and a
large number of combinations of parameters, we show
in ® gures 5± 7 only a selected set of results which should
demonstrate changes in correlation functions as the parameters vary, and also to what extent the theories are
able to reproduce the simulation data.
In ® gures 5 and 6 we show the results for the W3
model. The model has only two embedded sites which
do not seem, for smaller s R , to a€ ect the structure signi® cantly, and therefore the centre± centre correlation
function for s R = 0. 4 does not di€ er, qualitatively,
from the hard sphere correlation function. For this
value of s R the model strongly resembles a pseudo-molecular ¯ uid model [32], which is also seen from the
course of the remaining g i j ’s. Consequently, we also
see that the overall e€ ect of density is quite small. This
picture changes, however, when s R is increased. The
main changes are observed at lower densities, where
even g OO is not hard sphere-like any more. At higher
densities a restructuring takes place, as witnessed by a
qualitative change in g ++ ; only g O+ maintains its
pseudo-molecular character. This means that the e€ ect
of density for models with large s R is strong.
Considering the e€ ect of embedded sites, a model
similar to the W3 model is that of MeOH3. It has a
dumbbell core but only one embedded site and it can
thus be expected that its properties might copy those of
a corresponding pseudomolecular ¯ uid model based on
Fluids of pseudo-hard bodies. II
947
Figure 7. As for ® gure 5 for
the W5 model with s R =
0. 8 and at h = 0. 15 (left
column) and h = 0. 3
(right column).
hard dumbbells. Consequently, the e€ ect of the o€ -core
repulsions is really only marginal; regardless of density
and s R , and the ABF provides a reasonable zero order
approximation of g i j .
The above general picture, namely that for small s R
the e€ ect of o€ -core repulsions may be considered as a
small perturbation only, is valid for all models considerd. In this case all site± site correlation functions
can be estimated reasonably well using the concept of
a pseudo-molecular ¯ uid [32]. Consequently, they are
also predicted very well by the RAM theory in all
frames.
In ® gure 7 we show more complex behaviour exhibited by the model with predominance of the o€ -core
interactions, namely the W5 model with s R = 0. 8.
Very similar ® gures could be presented also for the
Amm5 and MeOH4 models. It is seen that the g OO is
not hard sphere-like even at higher densities. Further-
more, the g ++ correlation function (identical to g - - in
this case) exhibits a high peak at contact for higher
density, giving a substantial contribution to the overall
pressure. This is a direct consequence of a large number
of embedded sites and their repulsive interaction which
dwarfs the contribution of the core to the structure.
We have used all theories outlined in the preceding
section to calculate g i j ’s. These results are also shown for
comparison in ® gures 5± 7. From even a quick glance at
the ® gures it is immediately clear that the overall performance of all theories is rather poor and this applies
® rst of all to all versions of the OZ equation based
theories, whose performance seems unpredictable. In the
most important case of the centre± centre correlation
function, undoubtedly the RISM-PY version performs
best, and its results agree in all cases at least semi-qualitatively with the simulation data. This is also the only
case when the OZ equation based approach may be of
948
Fluids of pseudo-hard bodies. II
some relevance; in all other cases it is completely useless,
yielding unpredictable and often even unphysical results.
As regards the RAM theory, its overall performance
is much better than that of theories based on the OZ
equation. For obvious physical reasons (cf. discussion
above) it yields the best results for models with s R = 0. 4
at all densities and for models with s R = 0. 8 at lower
densities. In these cases the agreement with simulation
data is very good. At higher densities and for large s R ,
the predictions of the RAM theory are qualitatively
correct for the centre± centre functions, and semi-qualitative in the plus± minus frames. In the like-site frames,
the correlation functions predicted by the RAM theory
are too smooth to reproduce the observed extrema properly; nonetheless, they maintain at least approximately,
a correct qualitative shape.
4.
Conclusions
In the present paper we have examined theoretical
methods to estimate both the thermodynamic and structural properties of pseudo-hard body ¯ uids.
Concerning the equation of state, it seems di cult to
draw unique conclusion on applicability of di€ erent
methods. Although for some models the improved
scaled particle theory or a truncated perturbed virial
expansion performs well, this performance cannot be
always guaranteed and the methods may fail for other
models. A semi-empirical approach must therefore be
used if accurate and reliable results are required. A
one-parameter ® t has been found to perform excellently
for all models.
For the site± site correlation functions we used both
integral equations based on the Ornstein± Zernike equation ( RISM and RISM-like theories) and a perturbation
( RAM) theory. Despite an enormous e€ ort invested in
the implementation of the OZ equation based theories,
none of the versions (including new `corrected’ RISM)
performs satisfactorily. Indeed, in most cases the results
are even qualitatively wrong. Only the RISM-PY version seems to perform consistently for the centre± centre
correlation function. Concerning the RAM theory, it
performs comparably with or better than the RISMPY for g c c and maintains at least its qualitative correctness also for other g i j ’s.
To summarize, a ¯ uid of pseudo-hard bodies appears
to be too di cult for currently available theories and
their modi® cations to describe with at least reasonable
accuracy. This is very probably due to a non-additive
character of the site± site interactions which may be a
motivation for further fundamental theoretical research
of systems with complex interactions.
This work was supported by the Grant Agency of the
Czech Republic under Grant No. 203/96/0585 and by
the Grant Agency of the Academy of Sciences of the
Czech Republic under Grant No. A-4072607.
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]
[27]
[28]
[29]
[30]
[31]
[32]
References
Nez beda , I., 1997, Molec. Phys., 90, 661.
Nez beda , I., and Slovaík , J., 1997, Molec. Phys., 90, 353.
Wertheim, M. S., 1984, J. statist. Phys., 35, 19, 35.
Smith, W. R., and Nez beda , I., 1983, Adv. Chem. Ser.,
204, 235.
Kolafa , J., and Nez beda , I., 1987, Molec. Phys., 61,
161.
Walsh, J. M., Guedes, H. J. R., and Gubbins, K. E.,
1992, J. phys. Chem., 96, 10 995.
Nez beda , I., 1997, J. molec. L iquids, 73/74, 317.
Slovaík , J., and Nez beda , I., 1997, Molec. Phys., 91,
1125.
Nez beda , I., 1998, Czech. J. Phys. B, 48, 117.
Nez beda , I., P RÏedota , M., and A im, K., unpublished.
Gray, C. G., and Gubbins, K. E., 1984, Theory of
Molecular Fluids, Vol. 1 (Oxford: Clarendon Press).
A ndersen, H. C., and Chandler, D., 1972, J. chem.
Phys., 57, 1930.
Chandler, D., Silbey, and Ladanyi, B. M., 1982,
Molec. Phys., 46, 1335; Chandler, D., and Joslin,
C. G., 1982, Molec. Phys., 47, 871.
R ossky, P. J., and Chiles, R. A., 1984, Molec. Phys., 51,
661.
Jorgensen, W. L., 1981, J. Amer. chem. Soc., 103, 335.
Stillinger, F. H., and R ahman, A., 1974, J. chem.
Phys., 60, 1545.
Kolafa , J., and Nez beda , I., 1991, Molec. Phys., 72,
777.
Nez beda , I., Kolafa , J., and Smith, W. R., 1997, Fluid
Phase Equilibria, 130, 133.
Jorgensen, W. L., 1980, J. Amer. chem. Soc., 102,
543.
Jorgensen, W. L., and Ibrahim, M., 1980, J. Amer.
chem. Soc., 102, 3309.
Boubliík , T., and Nez beda , I., 1986, Coll. Czech. chem.
Commun., 51, 2301.
Nez beda , I., and Smith, W. R., 1981, Chem. Phys. L ett.,
81, 79.
Kalyuzhnyi, Y u , V., and Cummings, P. T., 1996,
J. chem. Phys., 105, 2011.
Kalyuzhnyi, Y u . V., and Cummings, P. T., 1996,
J. chem. Phys., 104, 3325.
Baxter, R. J., 1967, J. chem. Phys., 47, 4855.
Monson, P. A., 1984, Molec. Phys., 53, 1209.
A llen, M. P., and Tildesley, D. J., 1987, The Computer
Simulation of L iquids (Oxford: Clarendon Press).
Nez beda , J., and Kolafa , J., 1995, Molec. Simulation,
14, 153.
Labiík , S., and Nez beda , I., 1983, Molec. Phys., 48,
97.
Lado, F., 1983, Phys. Rev. A, 8, 2548; Malijevskyí, A.,
and Labik, S., 1987, Molec. Phys., 60, 663.
Labiík , S., Malijevskyí, A., and VoNÏka , P., 1985, Molec.
Phys., 56, 709.
Smith, W. R., Nez beda , I., and Labiík , S., 1984, J. chem.
Phys., 80, 5219.