Closed-loop liquid-liquid immiscibility in mixture of particles with spherically symmetric interaction
Thermodynamic perturbation theory for cetral-force (TPT-CF) type of associating potential is used to study the phase behavior of symmetric binary mixture of associating particles with spherically symmetric interaction. The model is represented by the binary Yukawa hard-sphere mixture with additional...
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irk-123456789-1208572017-06-14T03:04:28Z Closed-loop liquid-liquid immiscibility in mixture of particles with spherically symmetric interaction Kalyuzhnyi, Yu.V. Hvozd, T.V. Thermodynamic perturbation theory for cetral-force (TPT-CF) type of associating potential is used to study the phase behavior of symmetric binary mixture of associating particles with spherically symmetric interaction. The model is represented by the binary Yukawa hard-sphere mixture with additional spherically symmetric square-well associative interaction located inside the hard-core region and valid only between dissimilar species. To account for the change of the system packing fraction due to association we propose an extended version of the TPT-CF approach. In addition to already known four types of the phase diagram for binary mixtures we were able to identify the fifth type, which is characterized by the absence of the intersection of the λ-line with the liquid-vapour binodals and by the appearance of the closed- loop liquid-liquid immiscibility with upper and lower critical solution temperatures. В рамках термодинамiчної теорiї збурень для асоцiативного потенцiалу типу центральних сил проведено дослiдження фазової поведiнки симетричної бiнарної сумiшi асоцiативних частинок з сферично-симетричною взаємодiєю. Модель представлено бiнарною сумiшшю юкавiвських твердих сфер з додатковою сферично-симетричною асоцiативною взаємодiєю типу прямокутної ями, яка розмiщена усерединi областi твердої сфери i дiє тiльки мiж рiзними сортами. Враховуючи змiну упаковки системи внаслiдок асоцiацiї, запропоновано узагальнення термодинамiчної теорiї збурень для асоцiативного потенцiалу типу центральних сил. На додаток до чотирьох вже вiдомих типiв фазових дiаграм для бiнарних сумiшей, нам вдалося визначити п’ятий тип, який характеризується вiдсутнiстю перетину лямбда-лiнiї з бiнодалями “рiдина-газ” i появою незмiшування “рiдина-рiдина” у виглядi замкненої петлi з верхньою i нижньою критичними температурами змiшування. 2013 Article Closed-loop liquid-liquid immiscibility in mixture of particles with spherically symmetric interaction / Yu.V. Kalyuzhnyi, T.V. Hvozd // Condensed Matter Physics. — 2013. — Т. 16, № 4. — С. 43606:1-11. — Бібліогр.: 23 назв. — англ. 1607-324X PACS: 64.70.Ja, 05.70.Jk, 82.70.Dd, 64.10.+h DOI:10.5488/CMP.16.43606 arXiv:1312.4562 http://dspace.nbuv.gov.ua/handle/123456789/120857 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Thermodynamic perturbation theory for cetral-force (TPT-CF) type of associating potential is used to study the phase behavior of symmetric binary mixture of associating particles with spherically symmetric interaction. The model is represented by the binary Yukawa hard-sphere mixture with additional spherically symmetric square-well associative interaction located inside the hard-core region and valid only between dissimilar species. To account for the change of the system packing fraction due to association we propose an extended version of the TPT-CF approach. In addition to already known four types of the phase diagram for binary mixtures we were able to identify the fifth type, which is characterized by the absence of the intersection of the λ-line with the liquid-vapour binodals and by the appearance of the closed- loop liquid-liquid immiscibility with upper and lower critical solution temperatures. |
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Kalyuzhnyi, Yu.V. Hvozd, T.V. |
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Kalyuzhnyi, Yu.V. Hvozd, T.V. Closed-loop liquid-liquid immiscibility in mixture of particles with spherically symmetric interaction Condensed Matter Physics |
| author_facet |
Kalyuzhnyi, Yu.V. Hvozd, T.V. |
| author_sort |
Kalyuzhnyi, Yu.V. |
| title |
Closed-loop liquid-liquid immiscibility in mixture of particles with spherically symmetric interaction |
| title_short |
Closed-loop liquid-liquid immiscibility in mixture of particles with spherically symmetric interaction |
| title_full |
Closed-loop liquid-liquid immiscibility in mixture of particles with spherically symmetric interaction |
| title_fullStr |
Closed-loop liquid-liquid immiscibility in mixture of particles with spherically symmetric interaction |
| title_full_unstemmed |
Closed-loop liquid-liquid immiscibility in mixture of particles with spherically symmetric interaction |
| title_sort |
closed-loop liquid-liquid immiscibility in mixture of particles with spherically symmetric interaction |
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Інститут фізики конденсованих систем НАН України |
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2013 |
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http://dspace.nbuv.gov.ua/handle/123456789/120857 |
| citation_txt |
Closed-loop liquid-liquid immiscibility in mixture of particles with spherically symmetric interaction / Yu.V. Kalyuzhnyi, T.V. Hvozd // Condensed Matter Physics. — 2013. — Т. 16, № 4. — С. 43606:1-11. — Бібліогр.: 23 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT kalyuzhnyiyuv closedloopliquidliquidimmiscibilityinmixtureofparticleswithsphericallysymmetricinteraction AT hvozdtv closedloopliquidliquidimmiscibilityinmixtureofparticleswithsphericallysymmetricinteraction |
| first_indexed |
2025-07-08T18:44:31Z |
| last_indexed |
2025-07-08T18:44:31Z |
| _version_ |
1837105441948041216 |
| fulltext |
Condensed Matter Physics, 2013, Vol. 16, No 4, 43606: 1–11
DOI: 10.5488/CMP.16.43606
http://www.icmp.lviv.ua/journal
Closed-loop liquid-liquid immiscibility in mixture of
particles with spherically symmetric interaction
Yu.V. Kalyuzhnyi, T.V. Hvozd
Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine,
1 Svientsitskii St., 79011 Lviv, Ukraine
Received August 8, 2013, in final form September 9, 2013
Thermodynamic perturbation theory for central-force (TPT-CF) type of associating potential is used to study the
phase behavior of symmetric binary mixture of associating particles with spherically symmetric interaction. The
model is represented by the binary Yukawa hard-sphere mixture with additional spherically symmetric square-
well associative interaction located inside the hard-core region and valid only between dissimilar species. To
account for the change of the system packing fraction due to association we propose an extended version of
the TPT-CF approach. In addition to the already known four types of the phase diagram for binary mixtures
we were able to identify the fifth type, which is characterized by the absence of intersection of the ¸-line with
the liquid-vapour binodals and by the appearance of the closed- loop liquid-liquid immiscibility with upper and
lower critical solution temperatures.
Key words: thermodynamic perturbation theory, liquid-vapour coexistence, demixing, binary mixture,
associating fluids
PACS: 64.70.Ja, 05.70.Jk, 82.70.Dd, 64.10.+h
1. Introduction
According to the Gibbs phase rule, the binary mixture may have up to four coexisting phases simulta-
neously. This fact implies that the phase behavior of the binary fluid could be very rich and complicated.
Systematic study and classification of the peculiarities of the binary systems phase diagrams topologies
has been undertaken more than 40 years ago by Scott and van Konynenburg [1, 2]. These studies are
based on the application of the van der Waals equation of state, which in most cases is capable of provid-
ing qualitatively correct description of the phase behavior. Most of the subsequent studies, carried out
using quantitatively more accurate methods of the modern liquid state theory [3], have been focused on
the investigation of the phase behavior of symmetric binary mixtures [4–7]. These are the mixtures with
identical interaction between particles of the similar species and different interaction between particles
of the dissimilar species. Phase behavior of such mixtures is defined by the competition between gas-
liquid and liquid-liquid coexistence. Combining Monte-Carlo computer simulation and theoretical mean-
field calculations, Wilding et al. [4] identified three types of a phase diagram for square-well hard-sphere
symmetrical binary fluid mixture. Similar three types of a phase diagram were detected in the symmet-
rical binary hard-sphere Yukawa mixture using self-consistent Ornstein-Zernike approximation (SCOZA)
and Monte-Carlo computer simulation method [5, 6]. At the same time, the phase diagram of the fourth
type was also detected using SCOZA approach [8]. More recently, the first-order thermodynamic pertur-
bation theory has been used to study the phase behavior of the binary Yukawa mixture with asymmetry
in hard-sphere sizes [7].
In this study we are focused on the investigation of the phase behavior of symmetric Yukawa hard-
sphere binary mixture with additional spherically symmetric square-well associative interaction located
inside the hard-core region. This additional interaction is valid only between particles of dissimilar
species. The hard-sphere version of the model has been developed and studied by Cummings and Stell
© Yu.V. Kalyuzhnyi, T.V. Hvozd, 2013 43606-1
http://dx.doi.org/10.5488/CMP.16.43606
http://www.icmp.lviv.ua/journal
Yu.V. Kalyuzhnyi, T.V. Hvozd
[9] and by Kalyuzhnyi et al. [10]. Originally, this version of the model was used as a simple hamilto-
nian model of the chemical reaction [9]. On the other hand, the model of this type can be regarded as
a coarse grained version of the model for sterically or charge stabilized colloidal dispersions, protein
solutions, star-polymer fluids, etc. [11–13]. Effective interaction between macroparticles of such systems
has an attractive potential well at short distances and a repulsive potential mound at intermediate dis-
tances. The phase diagrams, which include two-phase gas-liquid, liquid-liquid diagrams and three-phase
gas-demixed liquid diagrams have been calculated using thermodynamic perturbation theory for central
force associating potential [14–16].
Phase behavior of the model, which is similar to the present one has been studied earlier by Jakson
[17]. His model is represented by the symmetrical binary hard-sphere mixture with mean-field type of
attractive interaction valid only between the same species and orientationally dependent associative in-
teraction between particles of unlike species. Associative interaction appears due to the off-center square-
well sites. This model was used as a generic model for the phase behavior description of the binary mix-
tures with the possibility of hydrogen bond formation between unlike species. Combining Wertheim’s
TPT for associating fluids and mean-field approach, Jackson was able to show that for a certain choice of
the potential model parameters, the system exhibits the closed loop liquid-liquid immiscibility with the
upper and lower critical solution temperatures. It was concluded that closed loop coexistence appeares
due to the presence of the highly anysotropic attraction between off-center bonding sites.
In the present workwe demonstrate that the systems with spherically symmetric interactionmay also
have closed loop liquid-liquid immiscibility. The paper is organized as follows. In section 2 we describe
the model to be considered and in section 3 we present and discuss details of the TPT-CF theory, special-
ized to the model at hand. Our results and discussion are included in section 4 and our conclusions are
collected in section 5.
2. The model
We consider symmetric binary Yukawa hard-sphere mixture with additional associative interaction
between dissimilar particles. The total pair potential of the modelU
i j
(r ) is represented as a sum of hard-
sphere Yukawa potentialUHSY
i j
(r ) and associative potentialUass(r ), i.e.:
U
i j
(r )ÆU
HSY
i j
(r )Å (1¡±
i j
)Uass(r ), (2.1)
where the lower indices i , j denote the species of the particles and ±
i j
is the Kroneker delta. In our
symmetric binary system, Yukawa interaction between particles of the same species is the same, i.e.,
U
HSY
11
(r ) ÆU
HSY
22
(r ), and between the particles of dissimilar species it is regulated by the parameter ®
(0Ç®Ç 1), i.e.UHSY
12
(r )ƮU
HSY
11
(r ). We have:
U
HSY
i i
(r )Æ
(
1, r É d
i i
,
¡
²
0
r
A
i i
z
n
e
¡z
n
(r¡d
i i
)
, r È d
i i
,
(2.2)
U
HSY
12
(r )ÆU
HSY
21
(r )Æ
8
<
:
1, r Ç L¡!/2 ,
A
0
, L¡!/2Ç r Ç d
12
,
¡
²
0
r
A
12
z
n
e
¡z
n
(r¡d
12
)
, r È d
12
,
(2.3)
where A
11
Æ A
22
Æ dz
n
, A
12
Æ A
21
ƮA
11
, z
n
and ²
0
are the screening length and the interaction strength
of the Yukawa potential, respectively, d
i j
Æ (d
i
Åd
j
)/2, d
i
is the hard-sphere diameter. We consider the
system with hard spheres of equal size, i.e., d
1
Æ d
2
Æ d . In (2.1)
Uass(r )Æ
8
<
:
0, r Ç L¡!/2 ,
¡"ass¡ A
0
, L¡!/2Ç r Ç LÅ!/2 ,
0, r È LÅ!/2 ,
(2.4)
where L is the bonding distance, ! and "ass are the square well potential width and depth, respec-
tively. In what follows we will consider the hard-sphere Yukawa potential (2.2) and (2.3) in the limit
of A
0
!1, and associative potential (2.4) in the limit of sticky interaction under the condition that the
43606-2
Closed-loop liquid-liquid immiscibility
second virial coefficient remains unchanged. In this limit, the Mayer function for associative potential
fass(r ) Æ exp[¡¯Uass(r )℄¡ 1 is substituted by the Dirac delta-function, i.e., e(HSY)
12
(r ) fass(r )! B±(r ¡ L),
where e
(HSY)
12
(r )Æ exp[¡¯U
(HSY)
12
(r )℄ and
B Æ L
¡2
Z
r
2
e
(HSY)
12
(r ) fass(r ) dr. (2.5)
The type of the clusters, which will be formed in the system due to association depends on the value
of the bonding length L [10, 14]. For values of L lying in the interval (0,d/2) only dimers can be formed
in the system. When d/2 Ç L Ç
p
3d , the formation of the chains is possible. Further increase in L leads
to an increase in the maximum number of the particles of the type 1 (or 2), which can be simultaneously
associated with the particle of the type 2 (or 1), and the formation of the branched chains will be possible.
The mixture is characterized by the temperature T (or ¯Æ (kBT )
¡1, where kB is the Boltzmann’s con-
stant), the total number-density ½, and the mole (number) fraction x of species 1 (x Æ x
1
); partial number
densities are defined via ½
1
Æ x½ and ½
2
Æ (1¡ x)½. We further introduce the reduced dimensionless
quantities, ½¤ Æ ½d3 , T ¤ Æ kBT /²
0
and ²¤ass Æ ²ass/²0.
3. Theory
To describe thermodynamic properties of the model at hand we will utilize here thermodynamic per-
turbation theory for central force associative potential (TPT-CF) [14–16]. According to TPT-CF, Helmholtz
free energy of the system A can be written as a sum of two terms: free energy of the reference system
Aref and the term describing the contribution to the free energy due to association Aass:
A Æ ArefÅ Aass Æ AHSYÅ Aass . (3.1)
Here, Aref Æ AHSY, where AHSY is the free energy of the hard-sphere Yukawa fluid. To calculate AHSY,
we are using the high temperature approximation. All the rest of thermodynamical quantities can be
obtained using the expression for Helmholtz free energy (3.1) and standard thermodynamical relations,
e.g., differentiating A with respect to the density, we get the expression for the chemical potential:
¯¹
k
Æ
�
�½
k
³
¯A
V
´
, (3.2)
and the expression for the pressure P of the system can be calculated invoking the following general
relation:
¯P Ư
X
k
½
k
¹
k
¡
¯A
V
. (3.3)
3.1. High temperature approximation
Under the high temperature approximation, the expression for the free energy is:
¯AHSY
V
Æ
¯AHS
V
Å2¼¯
X
i
X
j
½
i
½
j
1
Z
0
dr r
2
U
HSY
i j
(r )gHS(r ), (3.4)
where AHS is the hard-sphere Helmholtz free energy and gHS(r ) is the hard-sphere radial distribution
function. Substituting into (3.4) the expression for the potential (2.2) and (2.3), we have
¯AHSY
V
Æ
¯AHS
V
¡
2¼¯²
0
z
n
e
GHS(zn)
X
i
X
j
½
i
½
j
A
i j
, (3.5)
where e
GHS(zn) is the Laplace transform of hard-sphere radial distribution function
e
GHS(zn)Æ e
z
n
d
i j
1
Z
0
dr re
¡z
n
r
gHS(r ). (3.6)
43606-3
Yu.V. Kalyuzhnyi, T.V. Hvozd
Here, we will be using Percus-Yevick expression for e
GHS(zn), i.e.,
e
G
(HS)(zn)Æ
[¸
2
Å z
n
(¸
1
Ÿ
2
)℄
z
2
n
e
D
(n)
0
, (3.7)
where
¸
1
Æ¡
3
2
´
(1¡´)
2
, ¸
2
Æ
1Å2´
(1¡´)
2
, (3.8)
e
D
(n)
0
Æ
½
1¡12´
·
¸
1
Ÿ
2
z
2
n
³
1¡ z
n
d ¡e
¡z
n
d
´
Å
¸
2
z
3
n
³
1¡ z
n
d Å
z
2
n
2
¡e
¡z
n
d
´
¸¾
(3.9)
and ´Æ¼(½
1
Ž
2
)d
3
/6.
Differentiating the expression for Helmholtz free energy (3.4) with respect to the density, we get the
following expression for the chemical potential:
¯¹
(HSY)
k
Æ
�
�½
k
³
¯AHSY
V
´
Ư¹
(HS)
k
ů¢¹
(HSY)
k
, (3.10)
where ¹HSY
k
is the hard-sphere chemical potential and
¯¢¹
(HSY)
k
Æ¡
2¼¯²
0
z
n
"
�
e
GHS(zn)
�½
k
X
i
X
j
½
i
½
j
A
i j
Å2
e
GHS(zn)
X
i
½
i
A
ik
#
. (3.11)
Pressure PHSY of the system can be calculated invoking the following general relation:
¯PHSY Ư
X
k
½
k
¹
(HSY)
k
¡
¯AHSY
V
. (3.12)
In the above expressions, AHS and ¹
(HS)
k
are calculated using the corresponding Carnahan-Starling
expressions [18].
3.2. Thermodynamic perturbation theory
According to the TPT-CF for the associative part of the free energy Aass , we have:
¯Aass
V
Æ
X
k
"
½
k
ln
�
¾
(0)
k
½
k
!
Å
1
2
¾
(m¡1)
k
¾
(1)
k
¡¾
(0)
k
¾
(0)
k
#
, (3.13)
where
¾
(l )
k
ƾ
(0)
k
l
X
nÆ0
1
n!
�
¾
(1)
k
¡¾
(0)
k
¾
(0)
k
!
n
for l Æ 2, . . . ,m . (3.14)
Here,m is themaximumnumber of associative bonds per particle (themaximumnumber of the particles,
which can be bonded to a given particle simultaneously), ¾(l )
k
Æ
P
l
½
(l )
k
, ½
k
Æ
P
l
½
(l )
k
and ½(l )
k
is the density
of l -times bonded particles. For the present two-component mixture, the density parameters¾
(0)
k
and ¾
(1)
k
satisfy the following set of equations
8
>
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
>
:
¾
(1)
1
¡¾
(0)
1
¾
(0)
1
Æ
"
½
2
¡
1
m!
³
¾
(1)
2
¡¾
(0)
2
´
m
³
¾
(0)
2
´
m¡1
#
K ,
½
1
³
¾
(0)
1
´
m¡1
Æ
P
m
kÆ0
(¾
(0)
1
)
k
(m¡k)!
³
¾
(1)
1
¡¾
(0)
1
´
m¡k
,
¾
(1)
2
¡¾
(0)
2
¾
(0)
2
Æ
"
½
1
¡
1
m!
³
¾
(1)
1
¡¾
(0)
1
´
m
³
¾
(0)
1
´
m¡1
#
K ,
½
2
³
¾
(0)
2
´
m¡1
Æ
P
m
kÆ0
³
¾
(0)
2
´
k
(m¡k)!
³
¾
(1)
2
¡¾
(0)
2
´
m¡k
,
(3.15)
43606-4
Closed-loop liquid-liquid immiscibility
where
K Æ 4¼
Z
y
(00)
12
(r )e
(HSY)
(r ) fass(r )r
2
dr Æ 4¼BL
2
y
(00)
12
(L), (3.16)
y
(00)
12
(r ) represent the cavity distribution function between two Yukawa hard spheres of species 1 and 2
infinitely diluted in the original associating fluid in question. Usually, this function is approximated by
the hard-sphere Yukawa cavity correlation function y
(HSY)
12
(r,´) calculated for the packing fraction ´. This
appears to be a good approximation for the models with bonding length L ¼ d , since in this case y
(00)
12
(r )
only weakly depends on the degree of the system association. However, for L Ç d , the actual (effective)
packing fraction ´eff and thus y (00)
12
(r ) are strongly dependent on the system degree of association, and
the usual approximation becomes inadequate. In the present study we propose to approximate y
(00)
12
(r )
by the hard-sphere Yukawa cavity correlation function y
(HSY)
12
(r,´eff) calculated for the effective packing
fraction ´eff, i.e.,
´eff Æ
¼d
3
6
X
k
½
k
X
(0)
k
Å
m
X
nÆ1
µ
¼d
3
6
¡nVexc
¶
X
k
½
k
X
(0)
k
n!
�
X
(1)
k
X
(0)
k
!
n
, (3.17)
where the excluded volume Vexc is:
Vexc Æ
¼
24
(d ¡L)
2
(2d ÅL). (3.18)
X
(0)
k
Æ ½
(0)
k
/½
k
and X
(1)
k
Æ ½
(1)
k
/½
k
. According to this expression, ´eff [and thus y (HSY)
12
(r )] depends on the
degree of association of the system represented by the fractions of free X
(0)
k
and singly bonded X
(1)
k
par-
ticles.
In the present study, the solution of this equation is obtained via numerical iterationmethod. On each
iteration step, the new estimate for the fractions X (l )
k ,new
(l Æ 0,1) is calculated by solving the following set
of equations:
8
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
:
X
(1)
1,new
X
(0)
1,new
Æ
"
1¡
1
m!
³
X
(1)
2,new
´
m
³
X
(0)
2,new
´
m¡1
#
½
2
K
£
´eff(Xold)
¤
,
³
X
(0)
1,new
´
m¡1
Æ
P
m
nÆ0
³
X
(0)
1,new
´
n
(m¡n)!
³
X
(1)
1,new
´
m¡n
,
X
(1)
2,new
X
(0)
2,new
Æ
"
1¡
1
m!
³
X
(1)
1,new
´
m
³
X
(0)
1,new
´
m¡1
#
½
1
K
£
´eff(Xold)
¤
,
³
X
(0)
2,new
´
m¡1
Æ
P
m
nÆ0
³
X
(0)
2,new
´
n
(m¡n)!
³
X
(1)
2,new
´
m¡n
,
(3.19)
which is obtained using a set of equations (3.15). Here, Xold is the value of X calculated during the previ-
ous iteration step. Our iteration loop consists of two steps. In the first step, the current value of ´eff is used
to calculate the new values of X (l )
k
using the set of equation (3.19). On the second step, we insert these
values of X
(l )
k
into the right-hand side of the relation (3.17) to get a new estimate for ´eff. This iteration
loop is repeated until the following condition
j´eff,new¡´eff,oldj
´eff,newÅ´eff,old
É 10
¡8 (3.20)
is satisfied. For the initial guess we have used the value of ´eff Æ ´.
3.3. The cavity correlation function for Yukawa hard sphere fluid
The cavity correlation function y
(HSY)
12
(r ), which is needed to solve the set of equations for the fractions
X
(l )
k
(3.19) is calculated using the reference hypernetted chain type of approximation
y
(HSY)
i j
(r )Æ y
(HS)
i j
(r )exp
h
±h
(HSY)
i j
(r )¡±
(HSY)
i j
(r )
i
, (3.21)
43606-5
Yu.V. Kalyuzhnyi, T.V. Hvozd
where y
(HS)
i j
(r ) is the hard-sphere cavity correlation function, ±h
(HSY)
i j
(r ) Æ h
(HSY)
i j
(r ) ¡ h
(HS)
i j
(r ) and
±
(HSY)
i j
(r )Æ
(HSY)
i j
(r )¡
(HS)
i j
(r ). Here, the upper indices (HS) and (HSY) denote the hard-sphere and hard-
sphere Yukawa quantities, respectively, and h and
denote total and direct correlation functions, respec-
tively. In the hard-core region ±h
(HSY)
i j
(r )Æ 0 and for ±
(HSY)
i j
(r ), we have used the expression obtained in
the framework of the first-order mean spherical approximation [19]. The hard-sphere cavity correlation
function y
(HS)
i j
(r ) was calculated using Henderson-Grundke approximation [20]. Closed form analytical
expressions for ±
(HSY)
i j
(r ) and y
(HS)
i j
(r ) are presented in the appendix.
3.4. Calculation of the phase diagram
Our calculation of the phase diagram follows closely the scheme, proposed in [5]. It is based on the
solution of the set of equations that follow from the conditions of phase equilibrium, i.e., equal chemi-
cal potentials and pressures of the coexisting phases at a given temperature. The coexisting phases are
characterized by (½,x) and (½
0
,x
0
). From the Gibbs’ phase rule, we expect up to four phases to be in
equilibrium, i.e., the vapour (V), the mixed fluid (MF), and two (symmetric) phases of the demixed fluid
(DF).
The V-MF transition is obtained by solving the set of equations:
¹
i
(½,T,x Æ 1/2)´¹(½,T,x Æ 1/2)ƹ(½
0
,T,x Æ 1/2), (3.22)
P(½,T,x Æ 1/2)ÆP(½
0
,T,x Æ 1/2). (3.23)
The V-MF and MF-DF transitions are obtained in two steps: first we determine the phase diagram of the
demixing transitions , i.e., looking at a given temperature T for two coexisting states with the same fluid
density but different composition by fixing ½ Æ ½
0 and by determining concentrations x and x
0
Æ 1¡ x
of the coexisting phases. The equilibrium condition for the pressure is automatically fulfilled, while the
equilibrium condition for the chemical potentials takes place at given T and ½
¹(½,T,x)ƹ(½,T,x), (3.24)
which defines the line x(½) of the second order transition.
In the second step, the solution of the two equations
¹[½,T,x Æ 1/2℄ƹ[½
0
,T,x(½
0
)℄, (3.25)
P [½,T,x Æ 1/2℄ÆP [½
0
,T,x(½
0
)℄ (3.26)
gives the density ½ of the V or MF and the density of the DF with concentrations x(½0) and 1¡ x(½
0
), in
equilibrium.
4. Results and discussion
In this section we present our numerical results for the phase behavior of the model in question.
All the calculations are carried out at Yukawa screening parameter z
n
d Æ 1.8 and square-well width
!Æ 0.0000404981.
According to the previous studies [10], predictions of our theory for thermodynamical properties
of the model with A
i j
Æ 0 are in a good agreement with computer simulation predictions. To test the
accuracy of the theory for the model with ²
¤
ass Æ 0, we compare theoretical and computer simulation
predictions for its phase behavior. In figure 1 we show the phase diagram of the system at ²¤ass Æ 0 and
three values of ®, i.e., ® Æ 0.65, 0.7, 0.75. These are the system parameters for which the three types
of the phase diagram were identified [4, 5], depending on the position of intersection point of the ¸-
line, which represent the second-order demixing transition, with the binodals of the liquid-vapour (LV)
phase transition. In addition, for comparison in the same figure, we present the corresponding computer
simulations results [6]. Overall there is a reasonably good qualitative agreement between theoretical
and computer simulation predictions. Predictions of the theory in the region of the LV critical point are
43606-6
Closed-loop liquid-liquid immiscibility
Figure 1. Phase diagram of the symmetric binary Yukawa hard-sphere associating mixture in ½
¤ vs T¤
coordinate frame for ²¤ Æ 0 and ® Æ 0.65 (panel a), ® Æ 0.7 (panel b) and ® Æ 0.75 (panel c). Lines rep-
resent the results of the present theory and symbols depict MC computer simulation results [8]. Here,
dashed lines denote the ¸-lines and dotted lines show the metastable LV binodals.
about 6% higher than those of the computer simulation. As a result, while the types I and II of the phase
diagrams (according to the nomenclature of references [4, 5]) are theoretically reproduced for the set of
the potential model parameters used to simulate the type III of the diagram, theoretical calculations still
show the type II of the diagram with a small portion of stable binodals in the vicinity of the LV critical
point. However, it is quite obvious that a small decrease in ® will cause the theoretical phase diagram
to change its type from type II to type III. This can be seen in figure 2 (panel a), where our results for
® Æ 0.63 are shown. In the phase diagram of the type I the LV, coexistence is unstable with respect to
Figure 2. Phase diagram of the symmetric binary Yukawa hard-sphere associating mixture in ½
¤ vs T¤
coordinate frame (panel a) and in x vs T¤ frame for different values of the density ½¤ (panel b) at ²¤ Æ 0
and ®Æ 0.63. Dashed lines denote the ¸-line and solid and dotted lines represent stable and unstable por-
tions of the coexisting densities (panel a) and coexisting mole (number) fractions (panel b), respectively.
43606-7
Yu.V. Kalyuzhnyi, T.V. Hvozd
Figure 3. The same as in figure 2 at ²¤ Æ 5.2 and ®Æ 0.63.
the three-phase coexistence between mixed fluid (MF) and demixed fluid (DF) and the ¸-line ends at the
tricritical point. In the type II of the diagram, ¸-line ends also at the tricritical point, however, there is
a portion of the LV phase diagram beeing stable in the range of the temperatures between the critical
temperature and the temperature of the triple point, where one can observe LV coexistence at lower
densities. Between tricritical temperature and temperature of the triple point, the MF-DF three-phase
coexistence can be seen. At the triple point, there is a four-phase vapour, MF and DF coexistence. In the
case of the type II diagram, the ¸-line intersects the liquid binodal at the temperature slightly below the
critical. In the type III of the diagram, the ¸-line intersects the liquid binodal at the temperature well
below the critical temperature. Here, we can see the critical end point below which there is a three phase
V-DF coexistence and above which (up to the LV critical temperature), there is a LV coexistence. In the
type IV of the diagram, the ¸-line intersects LV binodals at the densities that are lower than the LV critical
densities [8]. This occures at ® Æ 0. We have also detected this type of the diagram, using the current
approach; however, the results are not shown here.
Next, we proceed to the discussion of the phase diagrams for the nonzero value of the strength of
associating interaction ²
¤
Æ 5.2, 6.0, 6.5 at ® Æ 0.63 (figures 3–5). Unfortunately, computer simulation
results for the phase behavior of the model at hand are not available. However, taking into account a
reasonable performance of the theory in the two limiting cases discussed above (A
i j
Æ 0 and ²
0
Æ 0), we
expect that the accuracy of the theory for the full version of the model will be satisfactory as well. In
figures 3–5, we depict the phase diagram in ½
¤ vs T ¤ (panel a) and x vs T ¤ at different values of the
density (panel b) frames. For ² Æ 0 and ® Æ 0.63, temperature-concentration slices of the phase diagram
at different densities are also shown (panel b in figure 2). In the latter case, only the upper portions of the
corresponding coexistence curves for ½¤ Æ 0.55, 0.6 are stable. The lower portions of these curves and the
demixing curves for ½¤ Æ 0.5, 0.45 are unstable with respect to the three-phase MF-DF coexistance. With
Figure 4. The same as in figure 2 at ²¤ Æ 6 and ®Æ 0.63.
43606-8
Closed-loop liquid-liquid immiscibility
Figure 5. The same as in figure 2 at ²¤ Æ 6.5 and ®Æ 0.63.
the temperature decrease, the difference in the compositions of the coexisting liquids increases. With the
increase of the strength of association ²¤, the topology of the phase behavior in T ¤ vs ½¤ coordinate frame
changes from the type III (figure 2) to type II at ²¤ Æ 5.2 (figure 3) and next to type I at ²¤ Æ 6.0 (figure 4). At
the same time, one can observe the appearance of the closed loop liquid-liquid immiscibility curves with
the upper stable and lower unstable critical solution points (figures 3 and 4). The stable portion of the
curves increases with an increasing strength of associative interaction. Finally, for ²¤ Æ 6.5, the closed-
loop coexistence curves for demixing coexistence becomes stable (figure 5, panel b). This corresponds
to the situation when there is no intersection between LV binodals a ¸-line (figure 5, panel a). Thus, for
these values of the potential model parameters, in addition to already identified four types of the phase
diagram, we have identified one more, which we call the type V of the two-component mixture phase
diagram topology. In a future we are planning to extend and apply our approach to study the effects of
the external field [21] and porous media [22, 23] on the phase behaviour of the current model.
5. Conclusions
In this paper we have used the TPT-CF approach to study the phase behavior of a symmetric two-
component Yukawamixture of associating particles with spherically symmetric interaction. Our theoret-
ical predictions for the phase diagram of the version of the model without association appear to be in
reasonably good qualitative agreement with the predictions of the corresponding Monte-Carlo computer
simulation method [6]. For the model with nonzero associating potential, we were able to identify, in
addition to the already known three types of the phase diagram topologies [4, 5], the type V of the phase
diagram. This type is characterized by the absence of intersection of the ¸-line, which represents a demix-
ing coexistence, with LV binodals. As a result, the stable closed-loop liqui-liquid immiscibility curves with
upper and lower critical solution temperatures can be observed for the larger values of the temperature
and density. Thus, closed-loop liquid-liquid immiscibility, which was observed earlier for the binary sys-
tems with highly directional attractive forces [17], can be also seen for the binary fluids with spherically
symmetric interaction.
A. Grundke-Henderson approximation
To calculate the hard-sphere cavity correlation function y
(HS)
i j
(r )we use Grundke-Henderson approx-
imation [20]. For r Ç d , we have:
ln y
(HS)
i j
(r )Æ
3
X
nÆ0
a
n
r
n
, (A.1)
43606-9
Yu.V. Kalyuzhnyi, T.V. Hvozd
where a
0
and a
1
are determined from (A.2) and (A.3), respectively, and a
2
and a
3
are determined by
requiring that y (HS)
i j
(r ) and �y (HS)
i j
(r )/�r should be continuous at r Æ d [(A.4)and (A.5)].
ln y
(HS)
i j
(0)Æ
8´¡9´
2
Å3´
3
(1¡´)
3
, (A.2)
³
� ln y
(HS)
i j
(r )
�r
´
(rÆ0)
Æ¡3´
2¡´
(1¡´)
3
, (A.3)
y
(HS)
i j
(d)Æ
4¡2´
4(1¡´)
3
, (A.4)
³
�y
(HS)
i j
(r )
�r
´
(rÆd)
Æ
5´
2
¡9/2´
(1¡´)
3
. (A.5)
Thus, we obtain a
0
, a
1
, a
2
, a
3
by combining equations (A.1), (A.2), (A.3), (A.4) and (A.5):
a
0
Æ
8´¡9´
2
Å3´
3
(1¡´)
3
, a
1
Æ¡3´
2¡´
(1¡´)
3
,
a
2
Æ
1
d
2
½
¡
(
3a
0
Å2a
1
d
)
Å3ln
·
4¡2´
4(1¡´)
3
¸
Å
9´Å10´
2
2¡´
¾
,
a
3
Æ
1
d
3
½
2a
0
Åa
1
d ¡2ln
·
4¡2´
4(1¡´)
3
¸
¡
9´Å10´
2
2¡´
¾
. (A.6)
B. First-order mean spherical approximation
Using first-order mean spherical approximation [19], for r Ç 1, we have:
±
(HSY)
i j
(r ) Æ ¯
²
0
r
A
i j
z
n
e
¡z
n
(r¡d)
¡
¯²
0
r
A
i j
z
n
1
(1¡´)
4
z
6
n
Q
2
0
(z
n
)
£
n
S
2
(z
n
)e
¡z
n
(r¡d)
Å144´
2
L
2
(z
n
)e
z
n
(r¡d)
Å24´S(z
n
)L(z
n
)
¡12´
2
£
(1Å2´)
2
z
4
n
Å (1¡´)(1Å2´)z
5
n
¤
r
4
Å12´
£
S(z
n
)L(z
n
)z
2
n
¡ (1¡´)
2
(1Å´/2)z
6
n
¤
r
2
¡24´
£
(1Å2´)
2
z
4
n
Å (1¡´)(1Å2´)z
5
n
¤
r
o
, (B.1)
where
Q
0
(z
n
)Æ
S(z
n
)Å12´L(z
n
)e
¡z
n
(1¡´)
2
z
3
n
, (B.2)
S(z
n
)Æ (1¡´)
2
z
3
n
Å6´(1¡´)z
2
n
Å18´
2
z
n
¡12´(1Å2´) , (B.3)
L(z
n
)Æ
³
1Å
´
2
´
z
n
Å1Å2´ . (B.4)
References
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Closed-loop liquid-liquid immiscibility
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Фазовий перехiд “рiдина-рiдина” iз замкнутою областю
незмiшування у сумiшi сферично-симетричних частинок
Ю.В. Калюжний, Т.В. Гвоздь
Iнститут фiзики конденсованих систем НАН України, вул. I. Свєнцiцького, 1, 79011 Львiв, Україна
В рамках термодинамiчної теорiї збурень для асоцiативного потенцiалу типу центральних сил прове-
дено дослiдження фазової поведiнки симетричної бiнарної сумiшi асоцiативних частинок з сферично-
симетричною взаємодiєю. Модель представлено бiнарною сумiшшю юкавiвських твердих сфер з дода-
тковою сферично-симетричною асоцiативною взаємодiєю типу прямокутної ями, яка розмiщена усере-
динi областi твердої сфери i дiє тiльки мiж рiзними сортами. Враховуючи змiну упаковки системи вна-
слiдок асоцiацiї, запропоновано узагальнення термодинамiчної теорiї збурень для асоцiативного потен-
цiалу типу центральних сил. На додаток до чотирьох вже вiдомих типiв фазових дiаграм для бiнарних
сумiшей, нам вдалося визначити п’ятий тип, який характеризується вiдсутнiстю перетину лямбда-лiнiї з
бiнодалями “рiдина-газ” i появою незмiшування “рiдина-рiдина” у виглядi замкненої петлi з верхньою i
нижньою критичними температурами змiшування.
Ключовi слова: термодинамiчна теорiя збурень, спiвiснування “рiдина-газ”, розшаровування, бiнарна
сумiш, асоцiативнi рiдини
43606-11
http://dx.doi.org/10.1063/1.1829632
http://dx.doi.org/10.1088/0953-8984/22/10/104113
http://dx.doi.org/10.1063/1.2042447
http://dx.doi.org/10.1080/00268978400100191
http://dx.doi.org/10.1063/1.468221
http://dx.doi.org/10.1038/nature03109
http://dx.doi.org/10.1529/biophysj.106.085142
http://dx.doi.org/10.1007/3-540-32702-9_13
http://dx.doi.org/10.1080/00268979300100821
http://dx.doi.org/10.1209/0295-5075/80/56002
http://dx.doi.org/10.5488/CMP.10.4.553
http://dx.doi.org/10.1080/00268979100100961
http://dx.doi.org/10.1063/1.1672048
http://dx.doi.org/10.1063/1.1541615
http://dx.doi.org/10.1063/1.431378
http://dx.doi.org/10.1209/epl/i2006-10087-7
http://dx.doi.org/10.1063/1.473042
http://dx.doi.org/10.1016/S0167-7322(03)00058-8
Introduction
The model
Theory
High temperature approximation
Thermodynamic perturbation theory
The cavity correlation function for Yukawa hard sphere fluid
Calculation of the phase diagram
Results and discussion
Conclusions
Grundke-Henderson approximation
First-order mean spherical approximation
|