Second-order Barker-Henderson perturbation theory for the phase behavior of polydisperse Morse hard-sphere mixture
We propose an extension of the second-order Barker-Henderson perturbation theory for polydisperse hard-sphere multi-Morse mixture. To verify the accuracy of the theory, we compare its predictions for the limiting case of monodisperse system, with predictions of the very accurate reference hypernette...
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| Cite this: | Second-order Barker-Henderson perturbation theory for the phase behavior of polydisperse Morse hard-sphere mixture / T.V. Hvozd, Yu.V. Kalyuzhnyi // Condensed Matter Physics. — 2015. — Т. 18, № 1. — С. 13605:1-13. — Бібліогр.: 13 назв. — англ. |
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Hvozd, T.V. Kalyuzhnyi, Yu.V. 2019-06-14T10:51:11Z 2019-06-14T10:51:11Z 2015 Second-order Barker-Henderson perturbation theory for the phase behavior of polydisperse Morse hard-sphere mixture / T.V. Hvozd, Yu.V. Kalyuzhnyi // Condensed Matter Physics. — 2015. — Т. 18, № 1. — С. 13605:1-13. — Бібліогр.: 13 назв. — англ. 1607-324X DOI:10.5488/CMP.18.13605 arXiv:1504.04255 PACS: 64.75.-g, 82.70.Dd https://nasplib.isofts.kiev.ua/handle/123456789/153539 We propose an extension of the second-order Barker-Henderson perturbation theory for polydisperse hard-sphere multi-Morse mixture. To verify the accuracy of the theory, we compare its predictions for the limiting case of monodisperse system, with predictions of the very accurate reference hypernetted chain approximation. The theory is used to describe the liquid--gas phase behavior of the mixture with different type and different degree of polydispersity. In addition to the regular liquid--gas critical point, we observe the appearance of the second critical point induced by polydispersity. With polydispersity increase, the two critical points merge and finally disappear. The corresponding cloud and shadow curves are represented by the closed curves with `liquid' and `gas' branches of the cloud curve almost coinciding for higher values of polydispersity. With a further increase of polydispersity, the cloud and shadow curves shrink and finally disappear. Our results are in agreement with the results of the previous studies carried out on the qualitative van der Waals level of description. Запропоновано застосування термодинамiчної теорiї збурень другого порядку Баркера-Хендерсона для дослiдження полiдисперсної сумiшi твердих сфер Морзе. Для перевiрки точностi порiвнюються результати цiєї теорiї для граничного випадку монодисперсної системи з результатами дуже точного базисного гiперланцюжкового наближення. Теорiя використовується для опису фазової поведiнки рiдина-газ для сумiшi з рiзними типами та рiзними ступенями полiдисперсностi. Окрiм звичайної критичної точки рiдина– газ, ми спостерiгаємо появу другої критичної точки, яка є зумовлена полiдисперснiстю. Iз збiльшенням полiдисперсностi цi двi критичнi точки зливаються i, нарештi, зникають. Вiдповiднi кривi хмари та тiнi представленi замкненими кривими з гiлками рiдина та газ, для кривої хмари вони майже збiгаються для вищих значень полiдисперсностi. При подальшому збiльшеннi полiдисперсностi кривi хмари та тiнi скорочуються i, нарештi, зникають. Нашi результати узгоджуються з результатами попереднiх дослiджень, якi були проведенi на якiсному рiвнi опису ван дер Ваальса. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Second-order Barker-Henderson perturbation theory for the phase behavior of polydisperse Morse hard-sphere mixture Застосування термодинамiчної теорiї збурень другого порядку Баркера-Хендерсона для дослiдження фазової поведiнки полiдисперсної сумiшi твердих сфер Морзе Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| title |
Second-order Barker-Henderson perturbation theory for the phase behavior of polydisperse Morse hard-sphere mixture |
| spellingShingle |
Second-order Barker-Henderson perturbation theory for the phase behavior of polydisperse Morse hard-sphere mixture Hvozd, T.V. Kalyuzhnyi, Yu.V. |
| title_short |
Second-order Barker-Henderson perturbation theory for the phase behavior of polydisperse Morse hard-sphere mixture |
| title_full |
Second-order Barker-Henderson perturbation theory for the phase behavior of polydisperse Morse hard-sphere mixture |
| title_fullStr |
Second-order Barker-Henderson perturbation theory for the phase behavior of polydisperse Morse hard-sphere mixture |
| title_full_unstemmed |
Second-order Barker-Henderson perturbation theory for the phase behavior of polydisperse Morse hard-sphere mixture |
| title_sort |
second-order barker-henderson perturbation theory for the phase behavior of polydisperse morse hard-sphere mixture |
| author |
Hvozd, T.V. Kalyuzhnyi, Yu.V. |
| author_facet |
Hvozd, T.V. Kalyuzhnyi, Yu.V. |
| publishDate |
2015 |
| language |
English |
| container_title |
Condensed Matter Physics |
| publisher |
Інститут фізики конденсованих систем НАН України |
| format |
Article |
| title_alt |
Застосування термодинамiчної теорiї збурень другого порядку Баркера-Хендерсона для дослiдження фазової поведiнки полiдисперсної сумiшi твердих сфер Морзе |
| description |
We propose an extension of the second-order Barker-Henderson perturbation theory for polydisperse hard-sphere multi-Morse mixture. To verify the accuracy of the theory, we compare its predictions for the limiting case of monodisperse system, with predictions of the very accurate reference hypernetted chain approximation. The theory is used to describe the liquid--gas phase behavior of the mixture with different type and different degree of polydispersity. In addition to the regular liquid--gas critical point, we observe the appearance of the second critical point induced by polydispersity. With polydispersity increase, the two critical points merge and finally disappear. The corresponding cloud and shadow curves are represented by the closed curves with `liquid' and `gas' branches of the cloud curve almost coinciding for higher values of polydispersity. With a further increase of polydispersity, the cloud and shadow curves shrink and finally disappear. Our results are in agreement with the results of the previous studies carried out on the qualitative van der Waals level of description.
Запропоновано застосування термодинамiчної теорiї збурень другого порядку Баркера-Хендерсона для
дослiдження полiдисперсної сумiшi твердих сфер Морзе. Для перевiрки точностi порiвнюються результати цiєї теорiї для граничного випадку монодисперсної системи з результатами дуже точного базисного
гiперланцюжкового наближення. Теорiя використовується для опису фазової поведiнки рiдина-газ для сумiшi з рiзними типами та рiзними ступенями полiдисперсностi. Окрiм звичайної критичної точки рiдина–
газ, ми спостерiгаємо появу другої критичної точки, яка є зумовлена полiдисперснiстю. Iз збiльшенням
полiдисперсностi цi двi критичнi точки зливаються i, нарештi, зникають. Вiдповiднi кривi хмари та тiнi
представленi замкненими кривими з гiлками рiдина та газ, для кривої хмари вони майже збiгаються для
вищих значень полiдисперсностi. При подальшому збiльшеннi полiдисперсностi кривi хмари та тiнi скорочуються i, нарештi, зникають. Нашi результати узгоджуються з результатами попереднiх дослiджень,
якi були проведенi на якiсному рiвнi опису ван дер Ваальса.
|
| issn |
1607-324X |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/153539 |
| citation_txt |
Second-order Barker-Henderson perturbation theory for the phase behavior of polydisperse Morse hard-sphere mixture / T.V. Hvozd, Yu.V. Kalyuzhnyi // Condensed Matter Physics. — 2015. — Т. 18, № 1. — С. 13605:1-13. — Бібліогр.: 13 назв. — англ. |
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2025-11-26T10:14:43Z |
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| fulltext |
Condensed Matter Physics, 2015, Vol. 18, No 1, 13605: 1–13
DOI: 10.5488/CMP.18.13605
http://www.icmp.lviv.ua/journal
Second-order Barker-Henderson perturbation theory
for the phase behavior of polydisperse Morse
hard-sphere mixture
T.V. Hvozd, Yu.V. Kalyuzhnyi
Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine,
1 Svientsitskii St., 79011 Lviv, Ukraine
March 25, 2015
We propose an extension of the second-order Barker-Henderson perturbation theory for polydisperse hard-
sphere multi-Morse mixture. To verify the accuracy of the theory, we compare its predictions for the limiting case
of monodisperse system, with predictions of the very accurate reference hypernetted chain approximation. The
theory is used to describe the liquid–gas phase behavior of the mixture with different type and different degree
of polydispersity. In addition to the regular liquid–gas critical point, we observe the appearance of the second
critical point induced by polydispersity. With polydispersity increase, the two critical points merge and finally
disappear. The corresponding cloud and shadow curves are represented by the closed curves with ‘liquid’ and
‘gas’ branches of the cloud curve almost coinciding for higher values of polydispersity. With a further increase
of polydispersity, the cloud and shadow curves shrink and finally disappear. Our results are in agreement with
the results of the previous studies carried out on the qualitative van der Waals level of description.
Key words: thermodynamic perturbation theory, polydispersity, phase coexistence, colloidal systems, Morse
potential
PACS: 64.75.-g, 82.70.Dd
1. Introduction
A vast majority of industrially important colloidal and polymeric materials are intrinsically polydis-
perse, i.e., each particle in the system is unique in size, charge, shape, chain length, etc. This feature
of colloidal and polymeric systems has a profound effect on their phase behavior and may cause the ap-
pearance of new phases and new phase transitions. In addition, the phenomena associated with the phase
behavior of polydisperse systems, such as fractionation, are also of technological relevance. Usually, the
theoretical methods used to study polydisperse systems treat them as a mixture of an infinite number
of components, each characterized by a continuous variable ξ distributed according to a certain distri-
bution function f (ξ), e.g., for a polydisperse hard-sphere fluid, the hard-sphere size σ is usually used as
such a variable, i.e., ξ= σ. The theoretical study of the phase behavior of such fluids, using the methods
of the modern liquid state theory, represents a nontrivial challenge [1]. The main obstacle for theoretical
description arises due to the fact that now one has to deal with an infinite number of equations for co-
existing phases. One of the possibilities to overcome this obstacle is to resort to the so-called truncatable
free energy (TFE) models and combine them with the possibility of their analytical description. TFE mod-
els are approximate schemes, where the thermodynamic properties can be expressed by a finite number
of generalized moments of the distribution function f (ξ). As a result, the formally infinite number of the
phase equilibrium condition equations can bemapped onto a set of a finite number of nonlinear algebraic
equations for these moments and solved using standard numerical methods. This route was recently un-
dertaken in a number of studies, i.e., phase behavior of Yukawa and charged hard-sphere polydisperse
mixtures were studied using the analytical solution of the mean spherical approximation (MSA) [2–5] and
high temperature approximation (HTA) [6]. More recently, HTA and dimer thermodynamic perturbation
© T.V. Hvozd, Yu.V. Kalyuzhnyi, 2015 13605-1
http://dx.doi.org/10.5488/CMP.18.13605
http://www.icmp.lviv.ua/journal
T.V. Hvozd, Yu.V. Kalyuzhnyi
theory (DTPT) for associating fluids in combination with polymer MSAwere used to investigate the phase
behavior of a polydisperse mixture of Yukawa chain molecules [7, 8]. While the approaches based on the
analytical solution of the MSA appear to be rather accurate, their application is restricted to the systems
with a factorized version of Yukawa interaction, i.e., the matrix of the coefficients describing the strength
of interaction is factorized into the product of two vectors. On the other hand, HTA based descriptions,
being less accurate, are much more flexible and can be applied to a much larger variety of the potential
models [6].
Attempting to improve the accuracy of the HTA based approaches, we present an extension and appli-
cation of the second-order Barker-Henderson perturbation theory for the phase behavior description of
polydisperse multi-Morse hard-sphere mixture. The paper is organized as follows: In section 2 we intro-
duce the model and in section 3 we present a corresponding extension of the BH2 theory. Our numerical
results for the phase behavior of the one-Morse version of the model are presented and discussed in sec-
tion 4, while in section 5 we collect our conclusions. In addition, we include an Appendix with explicit
analytical expressions for thermodynamical properties of themulti-Morse hard-sphere polydisperse mix-
ture in question.
2. The model
We consider the mixture with interparticle pair potential represented by the generalized multi-Morse
hard-sphere potential
UHSM(ξ,ξ′;r ) =
{
∞, r Éσ(ξ,ξ′),
−ǫ0
∑NM
n=1
∑M
m=1(−1)m−1 Anm(ξ)Anm(ξ′)e−zn [r−σ(ξ,ξ′)], r >σ(ξ,ξ′),
(2.1)
where ξ is the polydispersity attribute, i.e., a continuous version of the species index, σ(ξ) is the hard-
sphere diameter of the particle of species ξ, σ(ξ,ξ′) = [σ(ξ)+σ(ξ′)]/2, zn and ǫ0 are the the screening
length and the interaction strength of the Morse potential, respectively. The form suggested for the multi-
Morse potential (2.1) is similar to that used earlier for the multi-Yukawa potential [6]. This form is very
flexible and can be used to model a large variety of realistic potentials by an appropriate choice of the
coefficients Anm(ξ) and zn , e.g., in reference [6] it is used to mimic a polydisperse Lennard-Jones mixture.
Here, NM denotes the number of the Morse potential tails and M stands for the number of terms in the
sum for one Morse tail. Note that original Morse potential consists of two terms, i.e., one is attractive
and the other is repulsive. In our hard-sphere Morse potential repulsive term is substituted by the hard-
sphere term.
The mixture is characterized by the temperature T [or β = (kBT )−1, where kB is the Boltzmann’s
constant], the total number-density ρ, and by the distribution function f (ξ) [
∫
f (ξ)dξ= 1].
3. Theory
3.1. Barker-Henderson second-order perturbation theory
To describe thermodynamic properties of polydisperse Morse hard-sphere mixture, we use here the
Barker-Henderson second-order perturbation theory. According to this theory, Helmholtz free energy of
the system A can be written as a sum of three terms: free energy of the reference system (Aref) and the
two perturbation terms describing the contribution to the free energy due to Morse potential (A1, A2):
A = Aref+ A1 + A2 = AHS+ A1 + A2 . (3.1)
Here, Aref = AHS, where AHS is the free energy of the hard-sphere fluid. The first-order term is:
βA1
V
= 2πβ
∫
dξ
∫
dξ′ρ(ξ)ρ(ξ′)
∞∫
0
dr r 2UHSM(ξ,ξ′;r )g(HS)(ξ,ξ′;r ), (3.2)
13605-2
Second-order Barker-Henderson perturbation theory
where g(HS)(ξ,ξ′;r ) is the hard-sphere radial distribution function. For the second-order term, we used
the macroscopic compressibility approximation (MCA):
βA2
V
=−πβ2
∫
dξ
∫
dξ′ρ(ξ)ρ(ξ′)
∞∫
0
dr r 2
(
∂ρ
∂p
)
HS
[
UHSM(ξ,ξ′;r )
]2
g(HS)(ξ,ξ′;r ), (3.3)
where
(
∂ρ/∂p
)
HS
= κHS is the isothermal compressibility of the hard-sphere reference fluid, which is
obtained from the Carnahan-Starling equation and given by
κHS =
(1−η)4
1+4η+4η2 −4η3 +η4
, (3.4)
where the packing fraction η is defined as η= π
6
∫
dξρ(ξ)σ3(ξ). Substituting into (3.2) and (3.3) the expres-
sion for the potential (2.1), we have
βA1
V
= −2πβǫ0
∫
dξ
∫
dξ′ρ(ξ)ρ(ξ′)
NM∑
n=1
M∑
m=1
(−1)m−1 Anm(ξ)Anm(ξ′)
×
[
σ(ξ,ξ′)G̃(HS)(ξ,ξ′; zn)−
∂G̃(HS)(ξ,ξ′; zn )
∂zn
]
, (3.5)
βA2
V
= −πβ2ǫ2
0κ
HS
∫
dξ
∫
dξ′ρ(ξ)ρ(ξ′)
NM∑
n=1
M∑
m=1
[
Anm(ξ)Anm(ξ′)
]2
×
[
σ(ξ,ξ′)G̃(HS)(ξ,ξ′;2zn )−
∂G̃(HS)(ξ,ξ′;2zn )
∂(2zn)
]
, (3.6)
where G̃(HS)(ξ,ξ′; zn) is the Laplace transform of hard-sphere radial distribution function
G̃(HS)(ξ,ξ′; zn ) = eznσ(ξ,ξ′)
∞∫
0
dr r e−znr g(HS)(ξ,ξ′;r ). (3.7)
Here, we use Percus-Yevick approximation for the hard-sphere radial distribution function, since the
analytical expressions for its Laplace transform is known. All the rest 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:
βµ(ξ) =
δ
δρ(ξ)
(βA
V
)
, (3.8)
and the expression for the pressure P of the system can be calculated invoking the following general
relation:
βP =β
∫
dξρ(ξ)µ(ξ)−
βA
V
. (3.9)
In the above expressions, AHS and µ(HS)(ξ) are calculated using the corresponding Mansoori et al. expres-
sions [9].
Within the framework of the BH2 approach, the model in question belongs to the class of ‘truncat-
able free energy models’, i.e., the models possessing thermodynamical properties (Helmholtz free energy,
chemical potential, pressure) defined by a finite number of generalized moments. In this study, we have
the following moments:
ml =
∫
dξρ(ξ)ml (ξ) f (ξ), ml (ξ) =σl , (3.10)
13605-3
T.V. Hvozd, Yu.V. Kalyuzhnyi
m(n)
l
=
∫
dξρ(ξ)m(n)
l
(ξ) f (ξ), m(n)
l
(ξ) =σlϕ(zn ,σ), ϕ(zn ,σ) =
1
z2
n
(
1− znσ−e−znσ
)
, (3.11)
m̃(n)
l
=
∫
dξρ(ξ)m̃(n)
l
(ξ) f (ξ), m̃(n)
l
(ξ) =σlϕ(2zn ,σ), ϕ(2zn ,σ) =
1
(2zn)2
(
1−2znσ−e−2znσ
)
, (3.12)
ḿ(n)
l
=
∫
dξρ(ξ)ḿ(n)
l
(ξ) f (ξ), ḿ(n)
l
(ξ) =σl
kϕ1(zn ,σ), ϕ1(zn ,σ) = e−znσ, (3.13)
˜́m(n)
l
=
∫
dξρ(ξ)˜́m(n)
l
(ξ) f (ξ), ˜́m(n)
l
(ξ) =σl
kϕ1(2zn ,σ), ϕ1(2zn ,σ) = e−2znσ, (3.14)
m(nm)
l
=
∫
dξρ(ξ)m(nm)
l
(ξ) f (ξ), m(nm)
l
(ξ) =σl A(nm), (3.15)
m̃(nm)
l
=
∫
dξρ(ξ)m̃(nm)
l
(ξ) f (ξ), m̃(nm)
l
(ξ) =σl (A(nm))2. (3.16)
Closed form analytical expressions for thermodynamical properties (Helmholtz free energy, chemical
potential, pressure) in terms of the generalized moments (3.10)–(3.16) are presented in the Appendix.
3.2. Phase equilibrium conditions
The main obstacle in theoretical studies of the phase behavior in polydisperse systems arises due to
the fact that one has to deal with an infinite number of equations for coexisting phases. However, for the
present ‘truncatable free energy model’, these equations can be written as a finite number of equations
for the corresponding generalized moments of the distribution function f (ξ) [10].
We assume that at a certain temperature T , the system, which is characterized by the parent density
ρ(0) and parent-phase distribution function f (0)(ξ), separates into q daughter phases with the densities
ρ(1), ρ(2) , . . . , ρ(q) , and q daughter distributions f (1)(ξ), f (2)(ξ), . . . , f (q)(ξ). All phase equilibrium condi-
tions of the polydisperse system can be simply obtained by generalizing from the multicomponent case
via the prescription ρi =→ ρ f (ξ)dξ. Due to this substitution, thermodynamic properties (Helmholtz free
energy, chemical potential, pressure) become functionals of the distribution function f (ξ). Thermody-
namic conditions of phase equilibrium imply the equality of the pressures,
P (1)(T, [ f (1)(ξ)]) = P (2)(T, [ f (2)(ξ)]) = . . . = P (q)(T, [ f (q)(ξ)]), (3.17)
and of the chemical potentials for each species ξ.
µ(1)(ξ,T, [ f (1)(ξ)])= µ(2)(ξ,T, [ f (2)(ξ)]) = . . . =µ(q)(ξ,T, [ f (q)(ξ)]). (3.18)
The phase separation constrained by the conservation of the total number of particles of each species ξ,
f (0)(ξ) =
q∑
α=1
f (α)(ξ)x(α), (3.19)
where x(α) = N (α)/N (0) is the ratio of the total number of particles, N (α), in phase α to the total number
of particles in the parent phase N (0). The conservation of the total volume occupied by the parent phase:
υ0 =
q∑
α=1
υ(α)x(α), (3.20)
where υ(α) = 1/ρ(α) , (α= 1,2,3, . . ., q). Finally, the normalization of the f (α)(ξ)
∫
f (α)(ξ)dξ= 1 (3.21)
in equation (3.19) implies the conservation of the total number of particles:
1 =
q∑
α=1
x(α). (3.22)
13605-4
Second-order Barker-Henderson perturbation theory
In this study, we will consider a two-phase fractionation of polydisperse Morse hard-sphere mixture
(α= 1,2). We assume that thermodynamical properties of the model depend on K generalized moments
m0, m1, m2, . . . , mK , which are defined as follows:
mk = ρ
∫
mk (ξ) f (ξ)dξ, k , 0, (3.23)
and m0 = ρ. For the case of two-phase equilibrium (α = 1,2), the conditions (3.17)–(3.22) lead to the
following set of relations:
P (1)(T, {m(1)
k
})= P (2)(T, {m(2)
k
}), (3.24)
∫
f (α)(ξ)dξ= 1, for α= 1 or α= 2, (3.25)
and
m(1)
k
= m(1)
0
∫
m(1)
k
(ξ) f (0)(ξ)H(ξ,T,m(2)
0 , {m(1)}{m(0)})dξ, k , 0, (3.26)
where
H(ξ,T,m(2)
0 , {m(1)}{m(0)}) =
(
ρ(1) −ρ(2)
)
A12(ξ,T,m(2)
0 , {m(1)}{m(0)})
(
ρ(1)ρ(2)/ρ(0) −ρ(2)
)
+
(
ρ(1) −ρ(1)ρ(2)/ρ(0)
)
A12(ξ,T,m(2)
0 , {m(1)}{m(0)})
, (3.27)
A12(ξ,T, {m(1)}{m(2)}) =
ρ(2)
ρ(1)
exp
[
µ(2)
ex (ξ,T, {m(2)})−µ(1)
ex (ξ,T, {m(1)})
]
(3.28)
and µ(1)
ex is the value of the chemical potential in phase 1 in excess of its ideal gas value.
The solution of the set of equations (3.24)–(3.26) for a given temperature T , for the density of the par-
ent phase ρ(0), and for the parent species distribution function f (0)(ξ), gives us the coexisting densities
ρ(α) of the two daughter phases and the corresponding species distribution functions f (α)(ξ),α= 1,2. The
coexisting densities for different temperatures give us binodals, which are terminated at a temperature
for which the density of one of the phases is equal to the density ρ(0) of the parent phase. These termina-
tion points form the cloud and shadow coexisting curves that intersect at the critical point characterized
by the critical temperature Tcr and the critical density ρcr = ρ(1) = ρ(2) = ρ(0). The cloud and shadow
curves can be obtained as a special solution of the general coexisting problem, when the properties of
one phase are equal to the properties of the parent phase: assuming that the phase 2 is the cloud phase,
i.e., ρ(2) = ρ(0), and following the above scheme we will end up with the same set of equations, but with
ρ(2) and f (2)(ξ) substituted by ρ(0) and f (0)(ξ), respectively.
4. Results and discussion
In this sectionwe present numerical results for the phase behavior of polydisperseMorse hard-sphere
mixture. For the species distribution function f (ξ), we have chosen a log-normal distribution, i.e.,
f (LN)(ξ) =
I
p
2π ln I
exp
[
−
ln2(I 3/2ξ)
2ln I
]
, (4.1)
where I is the polydispersity index. Log-normal distribution frequently occurs at the colloidal and poly-
meric processing [1]. Note that in the monodisperse limit (I = 1), this distribution is represented by the
Dirac delta-function, δ(ξ−1). On the contrary, when I becomes very large (I ≫ 1), the above distribution
becomes very wide, increasing hereby the importance of the particles with a large value of ξ. All calcu-
lations were carried out for the one-Morse version of the pair potential (2.1), i.e., NM = 1, M = 1, with
13605-5
T.V. Hvozd, Yu.V. Kalyuzhnyi
z1 = 1.8σ0. We consider two types of polydispersity of the model: polydispersity only in the strength (am-
plitude) of the pair potential A11(ξ) and polydispersity in both, amplitude A11(ξ) and hard-sphere size
σ(ξ). In the former case, we have chosen A11(ξ) = A0ξ and σ(ξ) =σ0, while in the latter case A11(ξ) = A0ξ
and σ(ξ) = σ0ξ
1/4. Here, A0 = 1 and σ0 is the hard-sphere size for a monodisperse version of the model
at I = 1, which is used as a distance unit. In what follows the density ρ and temperature T are presented
in reduced units, i.e., ρ∗ = ρσ3
0 and T ∗ = kBT /ǫ0.
As a first step in our numerical study, we perform the calculation of thermodynamical properties
of the monodisperse version of the model, i.e., for I = 1 using the present version of the second-order
BH theory and the reference hypernetted chain approximation with the bridge function due to Verlet
[11, 12]. The latter theory is known for being very accurate in predicting the properties of simple fluids
[12]. The comparison of the results of both theories for the pressure and chemical potential (figure 1) at
three different temperatures demonstrate that BH theory is capable of giving relatively accurate results
at lower and intermediate densities. At higher values of the densities, the predictions of the BH approach
are a bit less accurate.
-1
0
1
2
3
4
5
6
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
βP
ρ∗
-8
-6
-4
-2
0
2
4
6
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
βµ
ρ∗
Figure 1. The pressure as a function of the density (the upper panel) and the chemical potential as a
function of the density (the lower panel) along three isotherms; the upper set of curves corresponds to
T∗ = 2.5, the intermediate set refers to T∗ = 2 and the upper set belongs to T∗ = 1.5. Crosses are RHNC
results, solid lines correspond to the Barker-Henderson second-order results.
Next, we perform the calculation of the phase diagram of a polydisperse version of the model at
different values of polydispersity index I . Our results for the model with polydispersity only in the am-
plitude of the potential are presented in figure 2 and for the model with polydispersity in both amplitude
and hard-sphere size are shown in figure 3. One can see that polydispersity has a profound effect on the
phase diagram. When I = 1, the phase diagram consists of the usual binodal ending in the critical point.
In this case, the cloud and shadow curves coincide, the critical point being located at their maximum.
For a polydisperse system (I , 1), for each parent phase density ρ(0), there is a different binodal. Each
binodal is truncated at a maximum temperature, with the corresponding densities, ρ(1) and ρ(2) lying
on the cloud and shadow curves, respectively. For a critical value of ρ(0), ρ(0) = ρcr, the corresponding
binodal passes through the intersection of the cloud and shadow curves. This occurs at T = Tcr and, since
ρcr = ρ(1) = ρ(2) = ρ(0), the point (ρcr,Tcr) is a critical point, where two coexisting phases, (1) and (2),
become identical. We are interested in the phase behavior of the system at relatively large values of poly-
13605-6
Second-order Barker-Henderson perturbation theory
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
0 0.2 0.4 0.6 0.8 1
Τ∗
ρ∗
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
0 0.2 0.4 0.6 0.8 1
Τ∗
ρ∗
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
0 0.2 0.4 0.6 0.8 1
Τ∗
ρ∗
Figure 2. Phase diagrams of the polydisperse Morse hard-sphere mixture with amplitude polydispersity
only in the (ρ∗,T∗) plane for three different values of the polydispersity index I , I = 1.02967 (the upper
panel), I = 1.034 (the intermediate panel) and I = 1.035 (the lower panel), obtained using the Barker-
Henderson second-order perturbation theory, which includes cloud (solid line) and shadow (dotted line)
curves, two critical points and critical binodals (dashed lines). Filled circle denotes the position of the
critical points.
dispersity. For a small polydispersity, the system has only one critical point [2–6], which originates from
the regular liquid–gas (LG) critical point of the corresponding monodisperse version of the system. With
the polydispersity increase, there appeares an additional critical point induced by polydispersity. This
effect on the qualitative van der Waals level of description has been observed by Bellier-Castella et al.
[10, 13]. The second critical point, which we denote as polydisperse (P) critical point, is located at larger
values of the density and at lower values of the temperature. It is present for both types of polydisper-
sity studied (figure 2 and 3). With polydispersity increase, both LG and P critical points move towards
each other and, for a certain limiting value of polydispersity, they merge. There are no critical points
above this limiting value (figure 2, lower panel). With a further increase of polydispersity and at lower
temperatures, we expect that the two-phase coexistence becomes unstable and there appeares a region
of three-phase coexistence. For relatively high values of polydispersity studied here we observe a rather
unusual shape for the cloud and shadow curves. For both types of polydispersity, they are represented
by closed curves of elipsoidal- and ∆-like shapes for the shadow curves seen in figure 2 and figure 3,
13605-7
T.V. Hvozd, Yu.V. Kalyuzhnyi
0
0.5
1
1.5
2
2.5
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
Τ∗
ρ∗
0
0.5
1
1.5
2
2.5
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
Τ∗
ρ∗
0
0.5
1
1.5
2
2.5
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
Τ∗
ρ∗
Figure 3. Phase diagrams of the polydisperse Morse hard-sphere mixturewith size and amplitude polydis-
persity in the (ρ∗,T∗) plane for three different values of the polydispersity index I , I = 1.035 (the upper
panel), I = 1.045 (the intermediate panel) and I = 1.055 (the lower panel), obtained using the Barker-
Henderson second-order perturbation theory, which includes cloud (solid line) and shadow (dotted line)
curves, two critical points and critical binodals (dashed lines). Filled circle denotes the position of the
critical points.
respectively, and closed curves of the linear shape for the cloud curves (figure 2, 3). In the latter case,
the ‘liquid’ and ‘gas’ branches of the cloud curves almost coincide for the larger polydispersity (figure 2
and 3, lower panels). With a further increase of polydispersity, the cloud and shadow curves shrink and
finally disappear.
Finally, in figure 4 we display distribution functions of the polydisperse Morse hard-sphere mixture
with size and amplitude polydispersity at two values of the temperature, which are higher and lower than
the second critical point temperature, i.e., T ∗ = 1.9 and T ∗ = 1.4, respectively. We present distribution
functions of the coexisting phases on the critical binodal at T ∗ = 1.4 (figure 4, upper panel) and on the
coexisting cloud and shadow phases at T ∗ = 1.4 (figure 4, intermediate panel), and T ∗ = 1.9 (figure 4,
lower panel). As usual [2–5], on the binodal the particles with larger values of ξ fractionate to the liquid
phase while particles with smaller values of ξ fractionate into the gas phase. Fractionation of the particles
to the shadow phases for the model at hand depends on the temperature. For the temperatures larger
than the temperature of polydispersity induced critical point, we observe fractionation of the usual type,
13605-8
Second-order Barker-Henderson perturbation theory
0
0.5
1
1.5
2
2.5
3
3.5
0 0.5 1 1.5 2 2.5 3
f(
ξ)
ξ
0
0.5
1
1.5
2
2.5
3
3.5
0 0.5 1 1.5 2 2.5 3
f(
ξ)
ξ
0
0.5
1
1.5
2
2.5
3
3.5
0 0.5 1 1.5 2 2.5 3
f(
ξ)
ξ
Figure 4. In the upper panel, we display parent f (0)(ξ) (solid line) and daughter f (1)(ξ) (corresponds to
gas phase, dashed line) and f (2)(ξ) (corresponds to liquid phase, dotted line) distribution functions for
critical binodal at I = 1.055 and T∗ = 1.4. In the intermediate panel, we show f (1)(ξ) (corresponds to gas
shadow phase, dashed line), f (2)(ξ) = f (0)(ξ) (corresponds to liquid and gas cloud phases, solid line) and
f (1)(ξ) (corresponds to liquid shadow phase, dotted line) distribution functions at I = 1.055 and T∗ = 1.4.
In the lower panel, we show the same as in intermediate panel at T∗ = 1.9.
i.e., the liquid shadow phase contains particles with ξ larger than those of the gas cloud phase while the
gas shadow phase contains particles with ξ smaller than those of the liquid cloud phase (figure 4, lower
panel). Note that distribution functions for the gas and liquid cloud phases are always the same and equal
to the distribution function of the parent phase. The situation changes for the temperatures lower than
the temperature of the second critical point (figure 4, intermediate panel). In this case, both liquid and
gas shadow phases contain particles with lower values of ξ than those of the liquid and gas cloud phases.
At the same time, the liquid shadow phase has particles of a larger value of ξ than the gas shadow phase.
This behavior of the model is related to the fact that for temperatures lower than the temperature of the
second critical point, both branches of the shadow curve are located to the left of both branches of the
cloud curve, i.e., the density of the shadow phases is always lower than the density of the cloud phases.
13605-9
T.V. Hvozd, Yu.V. Kalyuzhnyi
5. Conclusions
In this paper we present an extension of the second-order Barker-Henderson (BH2) perturbation the-
ory for a polydisperse hard-spheremulti-Morsemixture. To verify the accuracy of the theory, we compare
its predictions for the limiting case of a monodisperse system, with predictions of the very accurate ref-
erence hypernetted chain approximation. The theory is used to describe the liquid–gas phase behavior
of the mixture with different type and different degree of polydispersity. In agreement with the previous
study [10, 13], which was carried out using the qualitative van der Waals level of description, we observe
the appearance of the second critical point induced by polydispersity. With the polydispersity increase,
the two critical points merge and finally disappear, i.e., for polydispersity larger than a certain thresh-
old value, there are no critical points. The corresponding cloud and shadow curves are represented by
the closed curves with ‘liquid’ and ‘gas’ branches of the cloud curves that almost coincide with the poly-
dispersity increase. With a further increase of polydispersity, the cloud and shadow curves shrink and
finally disappear.
Appendix
Here, we present expressions for thermodynamical properties in terms of the moments (3.10)–(3.16).
We have:
βA1
V
= −2πβǫ0
{
Q (nm)
1 (zn)
z2
nD(n)
0 (zn)
−
∂
∂zn
[
Q (nm)
0 (zn)
z2
nD(n)
0 (zn)
]}
= −
2πβǫ0
z2
nD(n)
0 (zn)
{
Q (nm)
1 (zn)−
∂Q (nm)
0 (zn)
∂zn
+Q (nm)
0 (zn)
[
2
zn
+
1
D(n)
0 (zn)
∂D(n)
0 (zn)
∂zn
]}
, (A.1)
βA2
V
= −πβ2ǫ2
0κ
HS
{
Q̃ (nm)
1 (2zn )
z2
nD̃(n)
0 (2zn )
−
∂
∂(2zn)
[
Q̃ (nm)
0 (2zn)
(2zn )2D̃(n)
0 (2zn)
]}
= −
πβ2ǫ2
0κ
HS
4z2
n D̃(n)
0 (2zn)
{
Q̃ (nm)
1 (2zn)−
∂Q̃ (nm)
0 (2zn )
∂(2zn)
+Q̃ (nm)
0 (2zn )
[
1
zn
+
1
D̃(n)
0 (2zn)
∂D̃(n)
0 (2zn )
∂(2zn)
]}
, (A.2)
where
D(n)
0 (zn) =∆2 −
2π
zn
(
∆+
πm3
2
)(
m(n)
0 +
m2
2
)
−2π
{
∆m(n)
1 +
π
4
[
m(n)
2
(
m2 ++2m(n)
0
)
−2
(
m(n)
1
)2
]}
, (A.3)
∆= 1−
πm3
6
,
Q (nm)
0 (zn) =
{
∆(m(nm)
0 )2 +
π
2
[(
m3 + znm(n)
2
)(
m(nm)
0
)2
+
( zn
2
m2 + znm(n)
0
)(
m(nm)
1
)2
]
+ zn
(
∆−πm(n)
1
)
m(nm)
1 m(nm)
0
}
, (A.4)
Q (nm)
1 (zn) =
{
∆m(nm)
0 m(nm)
1 +
π
2
[(
m3 + znm(n)
2
)
m(nm)
0 m(nm)
1 +
( zn
2
m2 + znm(n)
0
)
m(nm)
1 m(nm)
2
]
+ zn
(
∆−πm(n)
1
)(
m(nm)
1
)2
}
, (A.5)
∂Q (nm)
0 (zn)
∂zn
=
π
2
[(
zn
∂m(n)
2
∂zn
+m(n)
2
)(
m(nm)
0
)2
+
(
m2
2
+ zn
∂m(n)
0
∂zn
+m(n)
0
)(
m(nm)
1
)2
]
+
(
∆−πm(n)
1 −πzn
∂m(n)
1
∂zn
)
m(nm)
1 m(nm)
0 , (A.6)
13605-10
Second-order Barker-Henderson perturbation theory
∂D(n)
0 (zn)
∂zn
=
2π
z2
n
(
∆+
πm3
2
)(
m(n)
0 +
m2
2
)
−
2π
zn
(
∆+
πm3
2
) ∂m(n)
0
∂zn
−2π
{
∆
∂m(n)
1
∂zn
+
π
4
[
∂m(n)
2
∂zn
(
m2 +2m(n)
0
)
+2m(n)
2
∂m(n)
0
∂zn
−4m(n)
1
∂m(n)
1
∂zn
]}
, (A.7)
m(n)
l
=
1
z2
n
(
ml − znml+1 −ḿ(n)
l
)
, m(n)
l
(ξ) =
1
z2
n
[
ml (ξ)− znml+1(ξ)−ḿ(n)
l
(ξ)
]
, (A.8)
∂m(n)
l
∂zn
=
1
z3
n
(
−2ml + znml+1 +2ḿ(n)
l
+ znḿ(n)
l+1
)
. (A.9)
Expressions for D̃(n)
0 (2zn), Q̃ (nm)
0 (2zn ), Q̃ (nm)
1 (2zn), ∂Q̃ (nm)
0 (2zn )/∂(2zn ), ∂D̃(n)
0 (2zn )/∂(2zn) in the second-
order term of Helmholtz free energy (A.2) is obtained replacing zn , m(n)
l
, ḿ(n)
l
, m(nm)
l
by 2zn , m̃(n)
l
, ˜́m(n)
l
,
m̃(nm)
l
, respectively, in expressions (A.3)–(A.9). Differentiating
βA1
V
with respect to the density we get the
expression for the chemical potential βµ1(ξ):
βµ1(ξ) =−
2πβǫ0
z2
nD(n)
0 (zn)
{
δQ (nm)
1 (zn)
δρ(ξ)
−
δ
δρ(ξ)
(
∂Q (nm)
0 (zn)
∂zn
)
+
(
δQ (nm)
0 (zn)
δρ(ξ)
−
Q (nm)
0 (zn)
D(n)
0 (zn)
×
δD(n)
0 (zn)
δρ(ξ)
)(
2
zn
+
1
D(n)
0 (zn)
∂D(n)
0 (zn)
∂zn
)
+
Q (nm)
0 (zn)
D(n)
0 (zn)
[
δ
δρ(ξ)
(
∂D(n)
0 (zn)
∂zn
)
−
1
D(n)
0 (zn)
×
δD(n)
0 (zn)
δρ(ξ)
∂D(n)
0 (zn)
∂zn
]
−
Q (nm)
1 (zn)
D(n)
0 (zn)
δD(n)
0 (zn)
δρ(ξ)
+
1
D(n)
0 (zn)
δD(n)
0 (zn)
δρ(ξ)
∂Q (nm)
0 (zn)
∂zn
}
, (A.10)
βµ2(ξ) =−
πβ2ǫ2
0
(2zn )2D̃(n)
0 (2zn)
(
κHS
{
δQ̃ (nm)
1 (2zn)
δρ(ξ)
−
δ
δρ(ξ)
(
∂Q̃ (nm)
0 (2zn)
∂(2zn )
)
+
(
δQ̃ (nm)
0 (2zn )
δρ(ξ)
−
Q̃ (nm)
0 (2zn)
D̃(n)
0 (2zn)
δD̃(n)
0 (2zn)
δρ(ξ)
)(
1
zn
+
1
D̃(n)
0 (2zn )
∂D̃(n)
0 (2zn)
∂(2zn)
)
+
Q̃ (nm)
0 (2zn)
D̃(n)
0 (2zn)
×
[
δ
δρ(ξ)
(
∂D̃(n)
0 (2zn )
∂(2zn)
)
−
1
D̃(n)
0 (2zn)
δD̃(n)
0 (2zn )
δρ(ξ)
∂D̃(n)
0 (2zn)
∂(2zn)
]
−
Q̃ (nm)
1 (2zn)
D̃(n)
0 (2zn)
δD̃(n)
0 (2zn)
δρ(ξ)
+
1
D̃(n)
0 (2zn )
δD̃(n)
0 (2zn)
δρ(ξ)
∂Q̃ (nm)
0 (2zn)
∂(2zn)
}
+
δκHS
δρ(ξ)
[
Q̃ (nm)
1 (2zn)−
∂Q̃ (nm)
0 (2zn)
∂(2zn)
+Q̃ (nm)
0 (2zn )
(
1
zn
+
1
D̃(n)
0 (2zn)
∂D̃(n)
0 (2zn )
∂(2zn )
)])
, (A.11)
where
δQ (nm)
0 (zn)
δρ(ξ)
=π
(
m3(ξ)
3
+
znm(n)
2 (ξ)
2
)(
m(nm)
0
)2
+
(
2∆+πm3 +πznm(n)
2
)
m(nm)
0 m(nm)
0 (ξ)
+πzn
[1
2
(
m2(ξ)
2
+m(n)
0 (ξ)
)(
m(nm)
1
)2
+
(m2
2
+m(n)
0
)
m(nm)
1 m(nm)
1 (ξ)
−
(
m3(ξ)
6
+m(n)
1 (ξ)
)
m(nm)
0 m(nm)
1 +
(
∆
π
−m(n)
1
)(
m(nm)
0 m(nm)
1 (ξ)+m(nm)
1 m(nm)
0 (ξ)
)]
,
(A.12)
13605-11
T.V. Hvozd, Yu.V. Kalyuzhnyi
δQ (nm)
1 (zn)
δρ(ξ)
=π
(
m3(ξ)
3
+
znm(n)
2 (ξ)
2
)
m(nm)
0 m(nm)
1 +
(
∆+
πm3
2
+
πznm(n)
2
2
)
×
(
m(nm)
0 m(nm)
1 (ξ)+m(nm)
1 m(nm)
0 (ξ)
)
+πzn
[1
2
(
m2(ξ)
2
+m(n)
0 (ξ)
)
m(nm)
1 m(nm)
2
+
1
2
(m2
2
+m(n)
0
)(
m(nm)
1 m(nm)
2 (ξ)+m(nm)
2 m(nm)
1 (ξ)
)
−
(
m3(ξ)
6
+m(n)
1 (ξ)
)
(m(nm)
1 )2
+
(
∆
π
−m(n)
1
)
2m(nm)
1 m(nm)
1 (ξ)
]
, (A.13)
δD(n)
0 (zn)
δρ(ξ)
=2π
{
1
3
πm3(ξ)
[
1
2
m(n)
1 −
1
zn
(
m(n)
0 +
1
2
m2
)]
−∆
(
1
6
m3(ξ)++m(n)
1 (ξ)
)
−
(
1
2
m2(ξ)+m(n)
0 (ξ)
)[ 1
zn
(
∆+
1
2
πm3
)
+
1
2
πm(n)
2
]
−
1
4
πm(n)
2 (ξ)
(
m2 +2m(n)
0
)
+πm(n)
1 m(n)
1 (ξ)
}
, (A.14)
δ
δρ(ξ)
(
∂Q (nm)
0 (zn)
∂zn
)
=
π
2
{[
zn
δ
δρ(ξ)
(
∂m(n)
2
∂zn
)
+m(n)
2 (ξ)
](
m(nm)
0
)2
+
(
zn
∂m(n)
2
∂zn
+m(n)
2
)
2m(nm)
0 m(nm)
0 (ξ)
+
[
m2(ξ)
2
+ zn
δ
δρ(ξ)
(
∂m(n)
0
∂zn
)
+m(n)
0 (ξ)
](
m(nm)
1
)2
+
(
m2
2
+ zn
∂m(n)
0
∂zn
+m(n)
0
)
2m(nm)
1 m(nm)
1 (ξ)
}
−
[
πm3(ξ)
6
+πm(n)
1 (ξ)+πzn
δ
δρ(ξ)
(
∂m(n)
1
∂zn
)]
m(nm)
1 m(nm)
0
+
(
∆−πm(n)
1 −πzn
∂m(n)
1
∂zn
)(
m(nm)
1 m(nm)
0 (ξ)+m(nm)
1 (ξ)m(nm)
0
)
, (A.15)
δ
δρ(ξ)
(
∂D(n)
0 (zn)
∂zn
)
=
2π
z2
n
[
πm3(ξ)
3
(
m(n)
0 +
m2
2
)
+
(
∆+
πm3
2
)(
m(n)
0 (ξ)+
m2(ξ)
2
)]
−
2π
zn
[
πm3(ξ)
3
∂m(n)
0
∂zn
+
(
∆+
πm3
2
) δ
δρ(ξ)
(
∂m(n)
0
∂zn
)]
−2π
{
−
πm3(ξ)
6
∂m(n)
1
∂zn
+∆
δ
δρ(ξ)
(
∂m(n)
1
∂zn
)
+
π
4
[
∂m(n)
2
∂zn
(
m2(ξ)+2m(n)
0 (ξ)
)
+
(
m2 +2m(n)
0
) δ
δρ(ξ)
(
∂m(n)
2
∂zn
)
+2m(n)
2 (ξ)
∂m(n)
0
∂zn
+2m(n)
2
δ
δρ(ξ)
(
∂m(n)
0
∂zn
)
−4m(n)
1 (ξ)
∂m(n)
1
∂zn
−4m(n)
1
δ
δρ(ξ)
(
∂m(n)
1
∂zn
)]}
, (A.16)
δ
δρ(ξ)
(
∂m(n)
l
∂zn
)
=
1
z3
n
(
−2ml (ξ)+ znml+1(ξ)+2ḿ(n)
l
(ξ)+ znḿ(n)
l+1
(ξ)
)
. (A.17)
δκHS
δρ(ξ)
=−
2πm3(ξ)
3
[
(1−η)3
1+4η+4η2 −4η3 +η4
+
(
1+2η−3η2 +η3
)
(1−η)4
(
1+4η+4η2 −4η3 +η4
)2
]
, (A.18)
Expressions for
δD̃(n)
0 (2zn)
δρ(ξ)
,
δQ̃ (nm)
0 (2zn)
δρ(ξ)
,
δQ̃ (nm)
1 (2zn)
δρ(ξ)
,
δ
δρ(ξ)
[
∂Q̃ (nm)
0 (2zn )
∂(2zn)
]
,
δ
δρ(ξ)
[
∂D̃(n)
0 (2zn)
∂(2zn)
]
in the second-order term for chemical potential (A.11) are obtained by replacing zn , m(n)
l
, m(n)
l
(ξ), ḿ(n)
l
,
ḿ(n)
l
(ξ), m(nm)
l
, m(nm)
l
(ξ) by 2zn , m̃(n)
l
, m̃(n)
l
(ξ), ˜́m(n)
l
, ˜́m(n)
l
(ξ), m̃(nm)
l
, m̃(nm)
l
(ξ), respectively, in expres-
sions (A.12)–(A.17).
13605-12
Second-order Barker-Henderson perturbation theory
References
1. Sollich P., J. Phys.: Condens. Matter, 2002, 14, R79; doi:10.1088/0953-8984/14/3/201.
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4. Kalyuzhnyi Y.V., Kahl G., Cummings P.T., Europhys. Lett., 2005, 72, 96; doi:10.1209/epl/i2005-10202-4.
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6. Kalyuzhnyi Y.V., Hlushak S.P., J. Chem. Phys., 2006, 125, 034501; doi:10.1063/1.2212419.
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9. Mansoori G.A., Carnahan N.F., Starling K.E., Leland T.W., J. Chem. Phys., 1971, 54, 1523; doi:10.1063/1.1675048.
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Застосування термодинамiчної теорiї збурень другого
порядку Баркера-Хендерсона для дослiдження фазової
поведiнки полiдисперсної сумiшi твердих сфер Морзе
Т.В. Гвоздь, Ю.В. Калюжний
Iнститут фiзики конденсованих систем НАН України, вул. I. Свєнцiцького, 1, 79011 Львiв, Україна
Запропоновано застосування термодинамiчної теорiї збурень другого порядку Баркера-Хендерсона для
дослiдження полiдисперсної сумiшi твердих сфер Морзе. Для перевiрки точностi порiвнюються резуль-
тати цiєї теорiї для граничного випадку монодисперсної системи з результатами дуже точного базисного
гiперланцюжкового наближення. Теорiя використовується для опису фазової поведiнки рiдина-газ для су-
мiшi з рiзними типами та рiзними ступенями полiдисперсностi. Окрiм звичайної критичної точки рiдина–
газ, ми спостерiгаємо появу другої критичної точки, яка є зумовлена полiдисперснiстю. Iз збiльшенням
полiдисперсностi цi двi критичнi точки зливаються i, нарештi, зникають. Вiдповiднi кривi хмари та тiнi
представленi замкненими кривими з гiлками рiдина та газ, для кривої хмари вони майже збiгаються для
вищих значень полiдисперсностi. При подальшому збiльшеннi полiдисперсностi кривi хмари та тiнi ско-
рочуються i, нарештi, зникають. Нашi результати узгоджуються з результатами попереднiх дослiджень,
якi були проведенi на якiсному рiвнi опису ван дер Ваальса.
Ключовi слова: термодинамiчна теорiя збурень, полiдисперснiсть, фазове спiвiснування, колоїднi
системи, потенцiал Морзе
13605-13
http://dx.doi.org/10.1088/0953-8984/14/3/201
http://dx.doi.org/10.1063/1.1607952
http://dx.doi.org/10.1063/1.1737291
http://dx.doi.org/10.1209/epl/i2005-10202-4
http://dx.doi.org/10.1063/1.2042347
http://dx.doi.org/10.1063/1.2212419
http://dx.doi.org/10.1016/j.cplett.2007.07.052
http://dx.doi.org/10.1063/1.3028044
http://dx.doi.org/10.1063/1.1675048
http://dx.doi.org/10.1063/1.1316007
http://dx.doi.org/10.1080/00268978000102671
http://dx.doi.org/10.1063/1.463379
http://dx.doi.org/10.1063/1.1386905
Introduction
The model
Theory
Barker-Henderson second-order perturbation theory
Phase equilibrium conditions
Results and discussion
Conclusions
|