Liquid-gas phase behavior of Stockmayer fluid with high dipolar moment
A multi-density version of the thermodynamic perturbation theory for associative fluids with central force associative potential is developed. The theory is used to describe the liquid-gas phase behavior of Stochmayer fluid with different values of the dipole moment ranging from low to high. Resul...
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| Zitieren: | Liquid-gas phase behavior of Stockmayer fluid with high dipolar moment / Yu.V. Kalyuzhnyi, I.A. Protsykevitch, P.T. Cummings // Condensed Matter Physics. — 2007. — Т. 10, № 4(52). — С. 553-562. — Бібліогр.: 22 назв. — англ. |
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irk-123456789-1189002017-06-02T03:04:14Z Liquid-gas phase behavior of Stockmayer fluid with high dipolar moment Kalyuzhnyi, Yu.V. Protsykevitch, I.A. Cummings, P.T. A multi-density version of the thermodynamic perturbation theory for associative fluids with central force associative potential is developed. The theory is used to describe the liquid-gas phase behavior of Stochmayer fluid with different values of the dipole moment ranging from low to high. Results of the theory are in a reasonable agreement with corresponding computer simulation results in a whole range of the dipole moment values studied. Contrary to previous computer simulation findings our calculations predict the occurrence of the liquid-gas phase coexistence for the soft-sphere dipole fluid model. Запропонований багатогустинний варiант термодинамiчної теорiї збурень для асоцiйованих рiдин з асоцiативним потенцiалом взаємодiї типу центральних сил. Розвинута теорiя використана для опису фазової поведiнки рiдина-газ моделi Штокмаєра з рiзними значеннями дипольного моменту, якi змiнюються вiд низького до високого. Результати теорiї є у задовiльнiй згодi з вiдповiдними результатами комп’ютерного моделювання для всiх значень дипольного моменту. На вiдмiну вiд розрахункiв, виконаних ранiше методами комп’ютерного моделювання, нашi розрахунки передбачають iснування фазової рiвноваги рiдина-газ в моделi м’яких дипольних сфер. 2007 Article Liquid-gas phase behavior of Stockmayer fluid with high dipolar moment / Yu.V. Kalyuzhnyi, I.A. Protsykevitch, P.T. Cummings // Condensed Matter Physics. — 2007. — Т. 10, № 4(52). — С. 553-562. — Бібліогр.: 22 назв. — англ. 1607-324X PACS: 64.10.+h, 64.70.Fx, 82.70.Dd DOI:10.5488/CMP.10.4.553 http://dspace.nbuv.gov.ua/handle/123456789/118900 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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| description |
A multi-density version of the thermodynamic perturbation theory for associative fluids with central force associative
potential is developed. The theory is used to describe the liquid-gas phase behavior of Stochmayer
fluid with different values of the dipole moment ranging from low to high. Results of the theory are in a reasonable
agreement with corresponding computer simulation results in a whole range of the dipole moment
values studied. Contrary to previous computer simulation findings our calculations predict the occurrence of
the liquid-gas phase coexistence for the soft-sphere dipole fluid model. |
| format |
Article |
| author |
Kalyuzhnyi, Yu.V. Protsykevitch, I.A. Cummings, P.T. |
| spellingShingle |
Kalyuzhnyi, Yu.V. Protsykevitch, I.A. Cummings, P.T. Liquid-gas phase behavior of Stockmayer fluid with high dipolar moment Condensed Matter Physics |
| author_facet |
Kalyuzhnyi, Yu.V. Protsykevitch, I.A. Cummings, P.T. |
| author_sort |
Kalyuzhnyi, Yu.V. |
| title |
Liquid-gas phase behavior of Stockmayer fluid with high dipolar moment |
| title_short |
Liquid-gas phase behavior of Stockmayer fluid with high dipolar moment |
| title_full |
Liquid-gas phase behavior of Stockmayer fluid with high dipolar moment |
| title_fullStr |
Liquid-gas phase behavior of Stockmayer fluid with high dipolar moment |
| title_full_unstemmed |
Liquid-gas phase behavior of Stockmayer fluid with high dipolar moment |
| title_sort |
liquid-gas phase behavior of stockmayer fluid with high dipolar moment |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2007 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/118900 |
| citation_txt |
Liquid-gas phase behavior of Stockmayer fluid with high dipolar moment / Yu.V. Kalyuzhnyi, I.A. Protsykevitch, P.T. Cummings // Condensed Matter Physics. — 2007. — Т. 10, № 4(52). — С. 553-562. — Бібліогр.: 22 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT kalyuzhnyiyuv liquidgasphasebehaviorofstockmayerfluidwithhighdipolarmoment AT protsykevitchia liquidgasphasebehaviorofstockmayerfluidwithhighdipolarmoment AT cummingspt liquidgasphasebehaviorofstockmayerfluidwithhighdipolarmoment |
| first_indexed |
2025-07-08T14:51:50Z |
| last_indexed |
2025-07-08T14:51:50Z |
| _version_ |
1837090799144140800 |
| fulltext |
Condensed Matter Physics 2007, Vol. 10, No 4(52), pp. 553–562
Liquid-gas phase behavior of Stockmayer fluid with
high dipolar moment
Yu.V.Kalyuzhnyi1, I.A.Protsykevitch1, P.T.Cummings2
1 Institute for Condensed Matter Physics, Svientsitskoho 1, 79011, Lviv, Ukraine
2 Department of Chemical Engineering, Vanderbilt University, Nashville, Tennessee 37235, and Nanomaterials
Theory Institute, Center for Nanophase Material Sciences, Oak Ridge National Laboratory, Oak Ridge,
Tennessee 37830
Received November 27, 2007
A multi-density version of the thermodynamic perturbation theory for associative fluids with central force as-
sociative potential is developed. The theory is used to describe the liquid-gas phase behavior of Stochmayer
fluid with different values of the dipole moment ranging from low to high. Results of the theory are in a rea-
sonable agreement with corresponding computer simulation results in a whole range of the dipole moment
values studied. Contrary to previous computer simulation findings our calculations predict the occurrence of
the liquid-gas phase coexistence for the soft-sphere dipole fluid model.
Key words: Stockmayer fluid, phase diagram, association, dipole moment
PACS: 64.10.+h, 64.70.Fx, 82.70.Dd
1. Introduction
Stockmayer fluid is a simple and convenient model, which is often used to study the properties of
the polar fluids [1] and ferrofluids [2–4], in particular their phase behavior. Perhaps first theoretical
predictions for the liquid-gas phase diagram of Stockmayer fluid was given by Stell, Rasaiah and
Narang (SRN) [5,6]. Their calculations were carried out using the Padé approximation based on the
perturbation theory due to Pople [7] and Stell et al. [5,6]. Comparison of the theoretical results with
corresponding computer simulation results, carried out later [8,9], shows that predictions of the
theory are accurate in the region of relatively low values of reduced dipole moment µ∗
d
2 . 2, where
µ∗
d = µd/
√
εLJσ3
LJ, µd is the dipole moment and σLJ and εLJ are the Lennard-Jones (LJ) potential
parameters. In the region of higher values of µ∗
d the theory becomes substantially less reliable.
According to computer simulation studies [10] in this region, strong dipole-dipole interaction causes
the particles to form chains in “nose-to-tail” configuration. Similar behavior has been also detected
for the dipolar hard-sphere fluid [11]. Thus, for the theory to be successful, adequate account for the
chain formation is crucial. Recently we have developed thermodynamic perturbation theory (TPT),
which explicitly takes into account formation of the chains due to strong dipole-dipole interaction
[12]. The theory is based on the multi-density approach for associating fluids with central force
type of associating potential, which was developed earlier by one of the authors [13]. In turn the
latter approach represents an appropriate reformulation of Wertheim’s multi-density theory for
associating fluids [14,15]. We refer to our version of the TPT as TPT central force (TPTCF).
TPTCF proposed in [12] utilizes three-density version of the multidensity formalism developed
earlier [13]. In this paper we propose extension of the TPTCF with the possibility of using an
arbitrary number of the densities. This enables the theory to be used in describing the network
forming fluids. We apply TPTCF to the study of the phase behavior of Stockmayer fluid with the
dipole moment µ∗
d from a wide range of values (1 6 µ∗
d
2 6 36) and compare our theoretical results
against currently available computer simulation results. We also consider the model of the dipole
fluid with the pair potential represented by Stockmayer potential with attractive part of the LJ
potential multiplied by the variable parameter λ. For λ = 1 this potential is identical to original
c© Yu.V.Kalyuzhnyi, I.A.Protsykevitch, P.T.Cummings 553
Yu.V.Kalyuzhnyi, I.A.Protsykevitch, P.T.Cummings
Stockmayer potential and for λ = 0 it reduces to soft-sphere dipole potential. This model was used
to study the effects of the dispersive interaction on the phase behavior of the dipolar fluids [10].
The paper is organized as follows. In section 2 we present details of the theory and in section 3
specify the potential model to be studied. In section 4 we discuss our choice of the reference system
and its description. Results and discussion are presented in section 5 and in section 6 we collect
our conclusions.
2. Thermodynamic perturbation theory for central force
associative potential
As in our earlier studies [13,12] we represent the pairwise total potential energy U(12) and
corresponding Mayer function f(12) as a sum of the reference Uref(12), fref(12) and associative
Uass(12), Fass(12) pieces, i.e.:
U(12) = Uref(12) + Uass(12), (1)
and
f(12) = fref(12) + Fass(12), (2)
where Fass(12) = eref(12)fass(12), eref(12) = exp [−βUref(12)], β = 1/kBT , f(12) = e(12) − 1, T is
the absolute temperature and kB is the Boltzmann’s constant. Here 1 and 2 denote positions and
orientations of the particles 1 and 2. Due to the decomposition of the Mayer function f(12) (2) we
will have the following diagrammatic expressions for the logarithm of the grand partition function
Ξ and for the one-point density ρ(1) in terms of the activity z:
ln Ξ is the sum of all topologically distinct simple connected diagrams consisting of black z̃ circles,
fref and Fass bonds. Each bonded pair of z̃ circles has either a fref or a Fass bond.
ρ(1) is the sum of all topologically distinct simple connected diagrams obtained from ln Ξ by
replacing in all possible ways one black z̃ circle by a white z̃(1) circle labeled 1.
Here z̃(i) = z exp [−βU(i)], i denotes position and orientation of the particle i, and U(i) is an
external field. For a uniform system z̃(1) ≡ z.
Following [12–14] we introduce the definition of the S-mer diagrams. These are the diagrams
consisting of S circles, which are connected by Fass bonds. The circles, which are incident with
more than m Fass bonds are called oversaturated circles. The set of all possible S-mer diagrams
can be constructed in three steps: (i) generating the subset of all possible connected diagrams with
only Fass bonds, (ii) inserting combined bond eref = fref + 1 between all pairs of circles which
are adjacent to the same oversaturated circle and are not connected by a direct Fass bond and
(iii) taking all ways of inserting a fref bond between the pairs of circles which were not connected
during the previous two steps. As a result the diagrams, which appear in ln Ξ and ρ(1), can be
expressed in terms of the S-mer diagrams:
ln Ξ is the sum of all topologically distinct simple connected diagrams consisting of S-mer dia-
grams with S = 1, . . . ,∞ and fref bonds between pairs of circles in distinct S-mer diagrams.
The procedure for obtaining the expression for ρ(1) from ln Ξ remains unchanged.
The diagrams appearing in the z̃ expansion of the singlet density ρ(1) can be classified with
respect to the number of Fass bonds associated with the labeled white z̃(1) circle. We denote the
sum of the diagrams with k (0 6 k < m) Fass bonds connected to the white z̃(1) circle as ρk(1) and
the sum of the diagrams with m or more Fass bonds connected to the white z̃(1) circle as ρm(1).
Thus for ρ(1) we have:
ρ(1) =
m
∑
k=0
ρk(1). (3)
554
Phase behavior of Stockmayer fluid
The process of switching from the activity to a density expansion proceeds in the same fashion as
in [13,12]. Analyzing the connectivity of the diagrams in ρ(1) at a white z̃(1) circle we have
ρ0(1) = z̃(1) exp [c0(1)] , (4)
ρk(1) = ρ0(1)
k
∑
n1,...,nk=0
k
∏
l=1
1
nl!
cnl
l , (5)
where
∑k
l=1 lnl = k,
cl(1) =
δc(0)
δσm−l(1)
, (6)
and
σl(1) =
l
∑
k=0
ρk(1). (7)
Here c(0) is the sum of all topologically distinct simple irreducible diagrams consisting of S-mer
diagrams with S = 1, . . . ,∞ and fref bonds between circles in distinct S-mers. All circles in c(0)
are black circles carrying the factor σm−l if the corresponding circle is incident with l 6 m Fass
bonds. The circles, which are incident with more than m Fass bonds carry the factor σ0.
Using the above introduced density parameters σk(1) the virial expansion for the pressure P
can be written in the following form:
βPV =
∫
[
ρ(1) −
m
∑
k=0
σm−k(1)ck(1)
]
d(1) + c(0), (8)
where V is the volume of the system. This expression satisfies the regular thermodynamic relation
β (∂P/∂µ) = ρ̄, where ρ̄ =
∫
ρ(1)d(1) and µ is the chemical potential. Corresponding expression
for Helmholtz free energy A is
βA =
∫
[
ρ(1) ln
ρ0(1)
Λ
+
m
∑
k=1
σm−k(1)ck(1)
]
d(1) − c(0), (9)
where Λ is the thermal wavelength. This expression is derived using the regular thermodynamic
expression for Helmholtz free energy A = Nµ − PV together with the relation
βNµ =
∫
ρ(1)
[
ln
ρ0(1)
Λ
− c0(1)
]
d(1), (10)
which follows from (4). For the excess Helmholtz free energy ∆A = A − Aref we have
β∆A =
∫
[
ρ(1) ln
ρ0(1)
ρ(1)
+
m
∑
k=1
σm−k(1)ck(1)
]
d(1) − ∆c(0), (11)
where N is the number of the particles in the system, ∆c(0) = c(0)
− c
(0)
ref , Aref and c
(0)
ref are the
corresponding quantities for the reference system. Ordering the virial expansion (11) with respect
to the number of associating bonds Fass and neglecting the terms with more than one associating
bond we have
∆c(0) =
1
2
∫
g
(2)
ref (12)σm−1(1)fass(12)σm−1(2)d(1)d(2), (12)
c1(1) =
∫
g
(2)
ref (12)fass(12)σm−1(2)d(2), (13)
and
ck(1) = 0, for k > 2, (14)
555
Yu.V.Kalyuzhnyi, I.A.Protsykevitch, P.T.Cummings
where g
(2)
ref (12) is the reference system pair distribution function. Equality (14) together with (5)
yields the following relation between the densities
ρk(1)
ρ0(1)
=
1
k!
[c1(1)]
k
=
1
k!
[
ρ1(1)
ρ0(1)
]k
. (15)
Set of the relations (12), (13) and (15) define all the quantities needed to calculate Helmholtz
free energy of the system (11), provided that the properties of the reference system are known. For
the uniform system, free energy expression (11) will be simplified
β∆A
V
= ρ ln
(
σ0
ρ
)
+
1
2
σm−1
σ1 − σ0
σ0
, (16)
where
σl = σ0
l
∑
k=0
1
k!
(
σ1 − σ0
σ0
)k
for l = 2, . . . ,m, (17)
σ0 and σ1 satisfy the following set of equations
σ1 − σ0
σ0
=
[
ρ −
1
m!
(σ1 − σ0)
m
σm−1
0
]
K, (18)
ρσm−1
0 =
m
∑
k=0
σk
0
(m − k)!
(σ1 − σ0)
m−k
σk
0 (19)
and
K =
∫
g
(2)
ref (12)fass(12)d(2). (20)
Expression (16) is derived combining (11), (12), (13) and (15).
3. The model
We consider the fluid with Stockmayer pair potential, which is represented by the sum of the
Lennard-Jones (LJ) potential ULJ(r) and dipole-dipole potential Udd(12)
U(12) = ULJ(r) + Udd(12), (21)
where
ULJ(r) = 4εLJ
[
(σLJ
r
)12
−
(σLJ
r
)6
]
(22)
and
Udd(12) = −
µ2
d
r3
[2 cos θ1 cos θ2 − sin θ1 sin θ2 cos (φ1 − φ2)] (23)
Here σLJ and εLJ are LJ potential size and energy parameters, respectively, µd is the dipole moment,
θ1 and θ2 denote the angles between the dipole vectors and vector, which joins the centers of the
two particles and φ1 and φ2 are the azimuthal angles about this vector.
4. Definition of the reference system and its properties
Following [12] we have
Uref(12) = ULJ(r) + sin2q θ1 sin2q θ2 Udd(12), (24)
Uass(12) =
(
1 − sin2q θ1 sin2q θ2
)
Udd(12). (25)
Here q plays a role of the potential splitting parameter, since at different values of q different
portions of the total potential will be included into its reference and associative parts. In the
556
Phase behavior of Stockmayer fluid
limiting case of q = 0 we have: Uass(12) = 0 and Uref(12) = U(12). Due to our previous study [12]
the optimal choice for this parameter is q = 3 and this value is used here. According to the above
choice of the reference system potential, the potential energy minima at “nose-to-tail” configuration
(θ1 = θ2 = 0, θ1 = θ2 = π) are included into the associative potential Uass(12). These minima are
responsible for the formation of the chains in the system.
According to SRN scheme [5,6], Helmholtz free energy of the reference system Aref can be
written using the Padé approximation
Aref = A
(0)
ref + (µ∗
d)
4 A
(2)
ref
1 − (µ∗
d)
2
A
(3)
ref /A
(2)
ref
, (26)
where µ∗
d = µd/
√
kBTσ3
LJ, A
(0)
ref is the LJ contribution to Helmholtz free energy, which was calcu-
lated using the equation of state for the LJ fluid proposed by Kolafa and Nezbeda [16],
A
(2)
ref
NkBT
= −
1
4
ρ
∫
g
(2)
LJ (r)dr
∫
[Uref(12)]
2
dΩ1dΩ2 , (27)
A
(3)
ref
NkBT
=
1
6
ρ2
∫
g
(3)
LJ (r12, r13, r23)dr2dr3
∫
Uref(12)Uref(13)Uref(23) dΩ1dΩ2dΩ3, (28)
where g
(2)
LJ (r) is the LJ fluid radial distribution function (RDF) and dΩ denotes integration over
the orientations. After carrying out the angular integration in (27) and (28), we have
A
(2)
ref
NkBT
= −
1
4
ρ
∫
w2(r)g
(2)
LJ (r)dr , (29)
A
(3)
ref
NkBT
=
ρ2
6
∫
w3(r12, r13, r23)g
(3)
LJ (r12, r13, r23)dr2dr3 , (30)
where expressions for w2(r) and w3(r12, r13, r23) are presented in the Appendix and for the triplet
distribution function g
(3)
LJ (r12, r13, r23) we have used superposition approximation, i.e.
g
(3)
LJ (r12, r13, r23) = g
(2)
LJ (r12)g
(2)
LJ (r13)g
(2)
LJ (r23) . (31)
Finally for the pair distribution function of the reference system gref(12), the following approx-
imation was utilized
g
(2)
ref (12) = y
(2)
LJ (r) exp (−βUref(12)) , (32)
where y
(2)
LJ (r) is the LJ fluid cavity distribution function. This function was calculated using so-
lution of the Ornstein-Zernike (OZ) equation supplemented by the closure conditions due to Duh
et al. [17–19].
5. Results and discussion
Thermodynamical properties of Stockmayer fluid was calculated using three-density version
(m = 2) of the TPTCF expression for excess Helmholtz free energy (16) and Padé expression for
the reference system Helmholtz free energy (26). RDF g
(2)
LJ (r), which is needed as an input in (16)
and (26), was calculated using OZ equation together with the closure relation suggested by Duh
et al. [17–19]. For the pressure P and chemical potential µ the standard thermodynamical relations
Z = ρ
∂
∂ρ
(
βA
N
)
, µN = A + PV (33)
have been utilized. Here Z = βPV/N . In what follows the dipole moment µd, the pressure P , the
temperature T and the density ρ of the system will be represented by the dimensionless quantities
µ∗
d = µd/
√
εLJσ3
LJ, P ∗ = PσLJ/εLJ, T ∗ = kBT/εLJ and ρ∗ = ρσ3
LJ, respectively.
557
Yu.V.Kalyuzhnyi, I.A.Protsykevitch, P.T.Cummings
ρ∗
T ∗
(a)
1
3
5
0.80.70.60.50.40.30.20.10
2.6
2.4
2.2
2
1.8
1.6
1.4
1.2
1
ρ∗
T ∗
(b)
2
4
6
0.80.70.60.50.40.30.20.10
3
2.5
2
1.5
1
ρ∗
T ∗ (c)
36
36
30
30
16
0.80.70.60.50.40.30.20.10
18
16
14
12
10
8
6
4
Figure 1. Liquid-gas phase diagram of Stockmayer fluid at different values of the dipole moment
µ∗
d in T ∗ vs ρ∗ coordinate frame. The values of the squared dipole moment µ∗
d
2 are placed
nearby the corresponding curves. Solid curves denote the results of the present theory, dashed
lines denote the results of SRN approach and empty circles denote computer simulation results
[8,9,20]. Critical points are denoted by filled symbols.
Predictions of the present theory for the liquid-gas phase behavior of Stockmayer fluid at
different values of the dipolar moment µd are shown in figure 1. Our results are compared with
computer simulation results [8,9,20] and the results of the SRN type of the theory, to which the
present approach will be reduced if in (24) and (25) one assumes the zero value for the splitting
parameter q, i.e. q = 0. For the low values of the dipole moment (µ∗
d
2 = 1÷ 2) one can see a good
agreement between theoretical and computer simulation predictions. With the dipole moment
increase, the accuracy of the SRN approach rapidly decreases; at high values of the dipole moment
(µ∗
d
2 = 16 ÷ 36) its predictions become unsatisfactory. In the whole range of the dipole moment
values studied our theory gives predictions, which are in reasonable agreement with computer
simulation predictions. In general, our results for the gas branch of the coexistence curve are
more accurate than those for the liquid branch. Predictions of the present theory for the critical
temperature T ∗
cr are in a good agreement with computer simulation predictions. At the same time
SRN approach gives slightly better predictions for T ∗
cr at the lowest values of the dipole moment
studied (µ∗
d
2 = 1 ÷ 2) and for the critical density ρ∗cr at low and intermediate values of the dipole
moment (µ∗
d
2 = 1 ÷ 6). Similarly, predictions of our theory for the vapor pressure, which together
with computer simulation predictions are presented in figure 2, are sufficiently accurate at all values
558
Phase behavior of Stockmayer fluid
of the dipole moment studied. At the same time SRN theory is accurate only at lower values of
the dipole moment (µ∗
d
2 = 1 ÷ 3). In all cases studied our theory gives more accurate results for
the critical pressure.
P ∗
T ∗
(a)
1
5
3
2.62.42.221.81.61.41.21
0.25
0.2
0.15
0.1
0.05
0
P ∗
T ∗
(b)
2
64
32.521.51
0.25
0.2
0.15
0.1
0.05
0
Figure 2. Vapor pressure along the coexistence curves at different values of the dipole moment
µ∗
d. The values of the squared dipole moment µ∗
d
2 are placed nearby corresponding curves. Solid
curves denote results of the present theory, dashed lines denote results of SRN approach and
empty circles denote computer simulation results [8,9]. Critical points are denoted by filled
symbols.
Finally, using the theory developed above for Stockmayer fluid, we study the phase behavior of
the model dipolar fluid with the following pair potential
Uλ(12) = 4εLJ
[
(σLJ
r
)12
− λ
(σLJ
r
)6
]
+ Udd(12), (34)
where 0 6 λ 6 1. For λ = 1 this potential coincides with Stockmayer potential and for λ = 0,
expression (34) represents a potential model for soft-sphere dipolar fluid. This model was used to
study the role of the dispersive interaction in the phase behavior of the dipolar fluids [10]. Ac-
cording to computer simulations studies, carried out in this work, there was no evidence of the
liquid-gas phase transition for λ < 0.3. However, recent computer simulation studies, carried out
for the dipolar hard-sphere fluid, present the evidence for the liquid-gas phase transition [21] with
the critical point located at low temperature and density. Since there is a strong similarity between
hard-sphere and present (λ = 0) soft-sphere dipolar models one may expect that liquid-gas phase
transition for the latter model exists and occurs at a temperatures lower than those studied in [10].
Thermodynamical properties of the model at hand were calculated using the scheme outlined above.
Since the introduction of λ modifies LJ part of the interaction in (34) we were not able to use the
equation of state due to Kolafa and Nezbeda [16]. The term A
(0)
ref , which appears in the expression
for Helmholtz free energy (26), was calculated using perturbation theory of Barker and Henderson
[22]. Our results for the phase behavior of the present model together with the corresponding re-
sults of computer simulation study [10] at µ∗
d
2 = 4 and different values of λ are shown in figures 3
and 4. Our predictions are in a reasonably good agreement with computer simulation predictions.
Especially accurate are theoretical results for the critical temperature at all values of λ studied
(figures 3 and 4a). Similar to the case of Stockmayer fluid, less accurate are the predictions for
559
Yu.V.Kalyuzhnyi, I.A.Protsykevitch, P.T.Cummings
ρ∗
T ∗
0
0.35
0.5
0.6
0.75
1
0.70.60.50.40.30.20.10
2.5
2
1.5
1
0.5
Figure 3. Liquid-gas phase diagram of Stockmayer fluid with variable dispersive interaction (34)
for µ∗
d
2 = 4 and different values of the switching parameter λ. The values of λ are placed nearby
the corresponding curves. Solid curves denote the results of the present theory and empty circles
denote computer simulation results [10]. Critical points are denoted by filled symbols.
(a)
λ
T ∗
cr
10.80.60.40.20
2.5
2
1.5
1
0.5
0
(b)
λ
ρ∗cr
10.80.60.40.20
0.3
0.25
0.2
0.15
0.1
0.05
0
Figure 4. Critical temperature (a) and critical density (b) vs λ. Solid curves denote results of
the present theory and empty circles denote computer simulation results [10].
the critical density (figures 3 and 4b) and for the liquid branch of the coexistence curve (figure 3).
Unlike computer simulation study [10] our study predicts the existence of the liquid-gas phase tran-
sition for λ < 0.3. In the limiting case of soft-sphere dipolar model (λ = 0) theoretical predictions
for the critical temperature and critical density are T ∗
cr = 0.404 and ρ∗ = 0.043, respectively.
560
Phase behavior of Stockmayer fluid
6. Conclusions
Three-density version of the thermodynamic perturbation theory for associating fluids with
central force associating potential developed earlier [12] is extended and a corresponding version
of the multi-density thermodynamic perturbation theory is proposed. This extension enables the
theory to be used in describing the network forming fluids. We apply the theory to the study of
the phase behavior of Stockmayer fluid with the dipole moment µ∗
d in a wide range of its values
(1 6 µ∗
d
2 6 36). Results of the theory are in a reasonable agreement with currently available
computer simulation results in the whole range of the dipole moment values studied. The theory
is also applied to the study of the phase behavior of Stockmayer fluid with additional parameter,
which makes a dispersive part of interaction variable [10]. According to our calculations the liquid-
gas phase equilibrium exists in the whole range of the values of this parameter, including the
limiting case of soft-sphere dipolar fluid model.
7. Appendix
Angular integration in (27) and (28) was carried out analytically using Maple computer algebra
package. The final result reads:
w2(r) =
A
r6
, w3(r12, r13, r23) =
(
3
r12r13r23
)3
B(θ1, θ2), (35)
where A = 17825792/225450225 and
B (θ1, θ2) =
85
∑
i=1
ki
li
cos
(
m
(1)
i θ1 + m
(2)
i θ2
)
. (36)
The values of the integer numbers ki, li and mi can be obtained from the authors upon request.
References
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14. Wertheim M.S., J. Stat. Phys., 1984, 35, 19;35.
15. Wertheim M.S., J. Stat. Phys., 1986, 42, 477;495.
16. Kolafa J., Nezbeda I., Fluid Phase. Equil., 1994, 100, 1.
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18. Duh D.-M., Haymet A.D.T., J. Chem. Phys., 1995, 103, 2625.
19. Duh D.-M., Henderson D., J. Chem. Phys., 1996, 104, 6742.
20. Bartke J., Hentschke R., Phys. Rev. E, 2007, 75, 061503.
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22. Barker J.A., Henderson D., Rev. Mod. Phys., 1975, 48, 587.
561
Yu.V.Kalyuzhnyi, I.A.Protsykevitch, P.T.Cummings
Фазова поведiнка рiдина-газ моделi Штокмаєра з високим
значенням дипольного моменту
Ю.В.Калюжний1, I.A.Процикевич1, П.T.Каммiнгс2
1 Iнститут фiзики конденсованих систем, Свєнцiцького 1, 79011, Львiв, Україна
2 Унiверситет Вандербiльда, Нешвiл, Теннесi 37235 та Нацiональна лабораторiя в Оук Рiджi, Оук Рiдж,
Теннесi 37830
Отримано 27 листопада 2007 р.
Запропонований багатогустинний варiант термодинамiчної теорiї збурень для асоцiйованих рiдин з
асоцiативним потенцiалом взаємодiї типу центральних сил. Розвинута теорiя використана для опи-
су фазової поведiнки рiдина-газ моделi Штокмаєра з рiзними значеннями дипольного моменту, якi
змiнюються вiд низького до високого. Результати теорiї є у задовiльнiй згодi з вiдповiдними резуль-
татами комп’ютерного моделювання для всiх значень дипольного моменту. На вiдмiну вiд розра-
хункiв, виконаних ранiше методами комп’ютерного моделювання, нашi розрахунки передбачають
iснування фазової рiвноваги рiдина-газ в моделi м’яких дипольних сфер.
Ключовi слова: рiдина Штокмаєра, фазова дiаграма, асоцiацiя, дипольний момент
PACS: 64.10.+h, 64.70.Fx, 82.70.Dd
562
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