Phase equilibria of size- and charge-asymmetric primitive models of ionic fluids: the method of collective variables
We study the phase diagrams of z:1 size-asymmetric primitive models (PMs) using a theory that exploits the method of collective variables. Based on the explicit expression for the relevant chemical potential (conjugate to the order parameter) which includes the correlation effects up to the third or...
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| Cite this: | Phase equilibria of size- and charge-asymmetric primitive models of ionic fluids: the method of collective variables / O.V. Patsahan, T.M. Patsahan // Condensed Matter Physics. — 2010. — Т. 13, № 2. — С. 23004: 1-10. — Бібліогр.: 32 назв. — англ. |
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Patsahan, O.V. Patsahan, T.M. 2012-04-08T15:35:10Z 2012-04-08T15:35:10Z 2010 Phase equilibria of size- and charge-asymmetric primitive models of ionic fluids: the method of collective variables / O.V. Patsahan, T.M. Patsahan // Condensed Matter Physics. — 2010. — Т. 13, № 2. — С. 23004: 1-10. — Бібліогр.: 32 назв. — англ. 1607-324X PACS: 05.70.Jk, 64.70.F-, 64.75.-g https://nasplib.isofts.kiev.ua/handle/123456789/32090 We study the phase diagrams of z:1 size-asymmetric primitive models (PMs) using a theory that exploits the method of collective variables. Based on the explicit expression for the relevant chemical potential (conjugate to the order parameter) which includes the correlation effects up to the third order we consider some particular cases of asymmetric PMs (1:1, 2:1 and 3:1 systems). Coexistence curves and critical parameters are calculated as functions of size ratio λ=σ+/σ-. Similar to simulations, our theory predicts the reduction of coexistence regions as well as a decrease of critical parameters Tc* and ρc* with an increase of size asymmetry. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Phase equilibria of size- and charge-asymmetric primitive models of ionic fluids: the method of collective variables Фазова рівновага примітивних моделей іонних плинів із асиметрією в розмірах і зарядах: метод колективних змінних Article published earlier |
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Phase equilibria of size- and charge-asymmetric primitive models of ionic fluids: the method of collective variables |
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Phase equilibria of size- and charge-asymmetric primitive models of ionic fluids: the method of collective variables Patsahan, O.V. Patsahan, T.M. |
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Phase equilibria of size- and charge-asymmetric primitive models of ionic fluids: the method of collective variables |
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Phase equilibria of size- and charge-asymmetric primitive models of ionic fluids: the method of collective variables |
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Phase equilibria of size- and charge-asymmetric primitive models of ionic fluids: the method of collective variables |
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Phase equilibria of size- and charge-asymmetric primitive models of ionic fluids: the method of collective variables |
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phase equilibria of size- and charge-asymmetric primitive models of ionic fluids: the method of collective variables |
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Patsahan, O.V. Patsahan, T.M. |
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Patsahan, O.V. Patsahan, T.M. |
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2010 |
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Condensed Matter Physics |
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Інститут фізики конденсованих систем НАН України |
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Article |
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Фазова рівновага примітивних моделей іонних плинів із асиметрією в розмірах і зарядах: метод колективних змінних |
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We study the phase diagrams of z:1 size-asymmetric primitive models (PMs) using a theory that exploits the method of collective variables. Based on the explicit expression for the relevant chemical potential (conjugate to the order parameter) which includes the correlation effects up to the third order we consider some particular cases of asymmetric PMs (1:1, 2:1 and 3:1 systems). Coexistence curves and critical parameters are calculated as functions of size ratio λ=σ+/σ-. Similar to simulations, our theory predicts the reduction of coexistence regions as well as a decrease of critical parameters Tc* and ρc* with an increase of size asymmetry.
|
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1607-324X |
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https://nasplib.isofts.kiev.ua/handle/123456789/32090 |
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Phase equilibria of size- and charge-asymmetric primitive models of ionic fluids: the method of collective variables / O.V. Patsahan, T.M. Patsahan // Condensed Matter Physics. — 2010. — Т. 13, № 2. — С. 23004: 1-10. — Бібліогр.: 32 назв. — англ. |
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AT patsahanov phaseequilibriaofsizeandchargeasymmetricprimitivemodelsofionicfluidsthemethodofcollectivevariables AT patsahantm phaseequilibriaofsizeandchargeasymmetricprimitivemodelsofionicfluidsthemethodofcollectivevariables AT patsahanov fazovarívnovagaprimítivnihmodeleiíonnihplinívízasimetríêûvrozmírahízarâdahmetodkolektivnihzmínnih AT patsahantm fazovarívnovagaprimítivnihmodeleiíonnihplinívízasimetríêûvrozmírahízarâdahmetodkolektivnihzmínnih |
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Condensed Matter Physics 2010, Vol. 13, No 2, 23004: 1–10
http://www.icmp.lviv.ua/journal
Phase equilibria of size- and charge-asymmetric
primitive models of ionic fluids: the method of
collective variables
O.V. Patsahan, T.M. Patsahan
Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine,
1 Svientsitskii Str., 79011 Lviv, Ukraine
Received March 22, 2010, in final form April 1, 2010
We study the phase diagrams of z:1 size-asymmetric primitive models (PMs) using a theory that exploits the
method of collective variables. Based on the explicit expression for the relevant chemical potential (conju-
gate to the order parameter) which includes the correlation effects up to the third order we consider some
particular cases of asymmetric PMs (1:1, 2:1 and 3:1 systems). Coexistence curves and critical parameters
are calculated as functions of size ratio λ = σ+/σ−. Similar to simulations, our theory predicts the reduc-
tion of coexistence regions as well as a decrease of critical parameters T ∗
c and ρ∗c with an increase of size
asymmetry.
Key words: size- and charge asymmetric primitive models, coexistence curves, gas-liquid critical point,
method of collective variables
PACS: 05.70.Jk, 64.70.F-, 64.75.-g
1. Introduction
The study of phase diagrams of ionic systems in which the phase separation is mainly driven
by electrostatic forces is of great fundamental interest. Electrolyte solutions, molten salts and ionic
liquids are examples of systems with strong Coulomb interactions. The simplest representation
of the systems dominated by Coulomb interactions is provided by the so-called primitive models
(PMs). In these models, the ionic fluid is described as an electroneutral mixture of charged hard
spheres immersed in a structureless dielectric continuum. A special case of the two-component
PM consisting of monovalent equisize hard spheres is called a restricted primitive model (RPM).
Because of its simplicity, the RPM has been extensively studied both theoretically and by simu-
lations. Early studies established [1] that the system has a low-temperature gas-liquid like phase
transition. A reasonable theoretical description of gas-liquid critical point of RPM was accompli-
shed at a mean-field (MF) level using integral equation methods [2] and Debye-Hückel (DH) theory
[3]. Over the years numerous simulation studies have been devoted to the location of critical point
but reliable estimations have been obtained only recently. Critical temperature is now known to
an excellent accuracy while critical density only to a moderate accuracy [4, 5].
From practical and fundamental points of view, asymmetric PMs are much more important than
perfectly symmetric systems. The investigation of phase behavior of PMs with size and charge
asymmetry was started more recently. The key findings from simulation studies of asymmetric
models are as follows: the suitably normalized critical temperatures decrease with size and charge
asymmetry while the critical densities increase with charge asymmetry but decrease with size
asymmetry [6–14]. Comparison of simulated critical parameters and theoretical predictions for
asymmetric models has revealed that several established theories, such as the mean spherical
approximation (MSA) and the original DH theory are not capable of predicting the trends observed
in simulations [15, 19]. Moreover, both the original DH theory and the MSA predict no dependence
on charge asymmetry in the equisize case. Only a direct inclusion of additional effects such as
association or geometrical constraints in the case of size asymmetry allowed one to qualitatively
c© O.V. Patsahan, T.M. Patsahan 23004-1
http://www.icmp.lviv.ua/journal
O.V. Patsahan, T.M. Patsahan
reproduce the trends found in simulations [18, 19]. It should be noted that most simulations have
concentrated on critical parameters and only a few studies [7, 8, 11] have addressed the phase
coexistence in asymmetric PMs.
Recently [20], we have proposed a method for the study of gas-liquid phase diagram of PMs
using the statistical field theory that exploits the method of collective variables (CVs) (see [21–24]).
The approach is based on determining the chemical potential conjugate to the order parameter
and enables one to take into account the effects of higher-order correlations. In [25], for the general
case of an asymmetric PM, we derived an analytical expression for the relevant chemical potential
which included the effects of correlations up to the third order. The results obtained for 1:1 and
2:1 systems qualitatively reproduced the dependence of critical temperature and critical density
on the size asymmetry. In this paper which is an extension of [25], we mainly focus on the effects
of size asymmetry on gas-liquid coexistence in asymmetric PMs.
The layout of the paper is as follows. In section 2 we outline the theory. We present the functional
of grand partition function of a size and charge-asymmetric PM in terms of CVs and determine
the order parameter. The method of calculating the relevant chemical potential is also briefly
described. In section 3 we calculate the gas-liquid coexistence data and the critical parameters for
some particular cases (1:1, 2:1 and 3:1 PMs). We conclude in section 4.
2. Theoretical background
2.1. Model
We consider a classical two-component system consisting of N+ cations carrying a charge q+ =
zq with diameter σ+ and N− anions carrying a charge q− = −q with diameter σ−. The ions are
immersed in a structureless dielectric medium. The system is electrically neutral:
∑
α=+,− qαρα = 0
and ρα = Nα/V is the number density of the αth species.
The pair interaction potential is assumed to be of the following form:
Uαβ(r) = φHS
αβ(r) + φC
αβ(r), (1)
where φHS
αβ(r) is the interaction potential between the two additive hard spheres of diameters σα and
σβ . We call the two-component hard sphere system a reference system (RS). Thermodynamic and
structural properties of RS are assumed to be known. φC
αβ(r) is the Coulomb potential: φC
αβ(r) =
qαqβ/(Dr), where D is dielectric constant, and hereafter we put D = 1. The model is characterized
by the size and charge asymmetry parameters:
λ =
σ+
σ−
, z =
q+
|q−|
. (2)
The fluid is at equilibrium in the grand canonical ensemble. The grand partition function of the
model (1) can be written as follows:
Ξ[να] =
∑
N+>0
∑
N
−
>0
∏
α=+,−
exp(ναNα)
Nα!
∫
(dΓ) exp
−β
2
∑
αβ
∑
ij
Uαβ(rij)
, (3)
where the following notations are used: να is dimensionless chemical potential, να = βµα −3 lnΛα,
µα is chemical potential of the αth species, β is reciprocal temperature, Λ−1
α = (2πmαβ−1/h2)1/2
is the inverse de Broglie thermal wavelength; (dΓ) is the element of configurational space of the
particles.
It is worth noting that regularization of the potential φC
αβ(r) inside the hard core is arbitrary
to some extent. For example, different regularizations for the Coulomb potential were considered
in [26, 27]. Within the framework of Gaussian approximation of the grand partition function, the
best estimation for critical temperature is achieved for optimized regularization [28] that leads to
an optimized random phase approximation. However, this approximation does not work properly
in higher orders of perturbation theory [27]. Here we use the Weeks-Chandler-Andersen (WCA)
regularization scheme for φC
αβ(r) [29].
23004-2
Phase equilibria of asymmetric primitive models
2.2. Functional integral. The Gaussian approximation
Using the CV based theory, developed in [24] for a multicomponent continuous system with
short- and long-range interactions in the grand canonical ensemble, we can write the functional
of the grand partition function of the PM (3) with the interaction potential (1) in the form of a
functional integral [25]:
Ξ[να] = ΞMF[ν̄α]
∫
(dρ)
∫
(dω) exp
{
− β
2V
∑
α,β
∑
k
φ̃C
αβ(k)ρk,αρ−k,β + i
∑
α
∑
k
ωk,αρk,α
+
∑
n>2
(−i)n
n!
∑
α1,...,αn
∑
k1,...,kn
Mα1...αn
(ν̄α; k1, . . . , kn)ωk1,α1
. . . ωkn,αn
δk1+···+kn
}
, (4)
where ΞMF[ν̄α] is the grand partition function of the model in the MF approximation and ν̄α is
the renormalized chemical potential
ν̄α = να +
β
2V
∑
k
φ̃C
αα(k).
Here ρk,α = ρc
k,α − iρs
k,α is the CV which describes the value of the k-th fluctuation mode of the
number density of the αth species, the indices c and s denote real and imaginary parts of ρk,α and
each of ρc
k,α (ρs
k,α) takes all the real values from −∞ to +∞. ωk,α is conjugate to the CV ρk,α.
(dρ) and (dω) are volume elements of the CV phase space
(dρ) =
∏
α
dρ0,α
∏
k6=0
′
dρc
k,αdρs
k,α , (dω) =
∏
α
dω0,α
∏
k6=0
′
dωc
k,αdωs
k,α ,
where the product over k is performed in the upper semi-space (ρ−k,α = ρ∗
k,α, ω−k,α = ω∗
k,α).
φ̃C
αβ(k) is the Fourier transform of the Coulomb potential φC
αβ(r). In the case of the WCA
regularization we obtain for βφ̃C
αβ(k)
βφ̃C
++(k) =
4πzσ3
±
T ∗(1 + δ)
sin(x(1 + δ))
x3
,
βφ̃C
−−(k) =
4πσ3
±
T ∗z(1 − δ)
sin(x(1 − δ))
x3
,
βφ̃C
+−(k) = −4πσ3
±
T ∗
sin(x)
x3
,
where the following notations are introduced:
T ∗ =
kBTσ±
q2z
(5)
is the dimensionless temperature, x = kσ±, σ± =
σ+ + σ−
2
and
δ =
λ − 1
λ + 1
. (6)
The nth cumulant Mα1...αn
coincides with the Fourier transform of the nth connected partial
correlation function of the RS [24]. Mα1...αn
depends on both the wave vectors ki and the partial
chemical potentials ν̄α.
Let us consider the Gaussian approximation of Ξ[να] setting Mα1...αn
≡ 0 for n > 3. Then,
after integration in (4) over ωk,α we obtain
ΞG[να] = ΞMF[ν̄α] Ξ′
∫
(dρ) exp
{
− 1
2N
∑
α,β
∑
k
C̃αβ(k)ρk,αρ−k,β
}
, (7)
23004-3
O.V. Patsahan, T.M. Patsahan
where C̃αβ(k) is the Fourier transform of the two-particle direct correlation function in the Gaussian
approximation
C̃αβ(k) = β
β
V
φ̃C
αβ(k) +
1
√
NαNβ
C̃HS
αβ (k),
C̃HS
αβ (k) is the Fourier transform of the direct correlation function of a two-component hard-sphere
system which is connected with Mαβ(k) by the relation C̃HS
2 (k)M2(k) = 1, where C̃HS
2 (k) denotes
the matrix of elements C̃HS
αβ (k)/
√
NαNβ and M2 the matrix of elements Mαβ(k); 1 is the unit
matrix.
The diagonalization of the square form in (7) leads us to the equation
ΞG[να] = ΞHS[ν̄α] Ξ′
∫
(dξ) exp
{
− 1
2
∑
α=1,2
∑
k
ε̃α(k)ξk,αξ−k,α
}
. (8)
Eigenvalues ε̃1(k) and ε̃2(k) in the long-wavelength limit are found to be [25]
ε̃1(k = 0) = ∞, (9)
ε̃2(k = 0) =
1 + z
1 + z2
(
− 4πρ∗zδ2
3T ∗(1 + z)
+ c̃HS
++(0) + 2
√
zc̃HS
+−(0) + zc̃HS
−−(0)
)
, (10)
where T ∗ and δ are given by (5)–(6) and
ρ∗ = ρσ3
± (11)
is a reduced total number density. Equation (9) leads to G̃QQ(k = 0) = 0, where G̃QQ(k = 0) is
the Fourier transformation of the charge-charge connected correlation function. It reflects the fact
that the first moment Stillinger-Lovett rule is satisfied.
It is worth noting that equation ε̃2(k = 0) = 0 (see (10)) leads us to the same expression for
gas-liquid spinodal as that obtained in [30] but for another type of regularization of the Coulomb
potential inside the hard core.
Eigenvectors ξk,1 and ξk,2 in the long-wavelength limit have the form [25]:
ξ0,1 =
1√
1 + z2
ρ0,Q,
ξ0,2 =
1√
1 + z2
(
1 + z2
1 + z
ρ0,N +
1 − z
1 + z
ρ0,Q
)
, (12)
where CVs ρ0,N = δρ0,++δρ0,− and ρ0,Q = zδρ0,+−δρ0,− describe long-wavelength fluctuations of
the total number density and charge density, respectively. As is seen, CV ξ0,1 describes fluctuations
of the charge density. In the general case z 6= 1, ξ0,2 is a linear combination of CVs ρ0,N and ρ0,Q
with the z-dependent coefficients. At z = 1, CV ξ0,2 solely describes fluctuations of the total
number density. Thus, we suggest that CV ξ0,2 is connected with the order parameter of the
gas-liquid critical point.
Finally, after integration in (8) we arrive at the logarithm of the grand partition function in
the Gaussian approximation
ln ΞG[να] = ln ΞHS[ν̄α] − 1
2
∑
k
ln det [1 + ΦCM2] , (13)
where ΦC and M2 are matrices of elements βφ̃C
αβ(k) and Mαβ(ν̄α; k), respectively.
23004-4
Phase equilibria of asymmetric primitive models
2.3. Beyond the Gaussian approximation
We study the gas-liquid phase diagrams of asymmetric PMs using the method proposed in [20].
First, we pass from the initial chemical potentials ν+ and ν− to their linear combinations
ν1 =
zν+ − ν−√
1 + z2
, ν2 =
ν+ + zν−√
1 + z2
.
Chemical potentials ν1 and ν2 are conjugate to CVs ξ0,1 and ξ0,2, respectively. Since we suggest
that CV ξ0,2 (see (12)) is connected with the order parameter, ν2 appears to be of special interest
in our study. It can easily be shown that ν2 is an electrochemical potential.
We start with the grand partition function in the Gaussian approximation (13) replacing the
cumulants Mαβ(ν̄α; k) by their values in the long-wavelength limit Mαβ(k) = Mαβ(k = 0) = Mαβ .
Taking into account that ln ΞHS and Mα1α2...αn
are functions of full chemical potentials ν1 and ν2,
we present ν1 and ν2 as
ν1 = ν0
1 + ∆ν1 , ν2 = ν0
2 + ∆ν2 ,
with ν0
1 and ν0
2 being the MF values of ν1 and ν2, respectively. Then, we self-consistently solve
equations
∂ ln ΞG(ν1, ν2)
∂∆ν1
= 0, (14)
∂ ln ΞG(ν1, ν2)
∂∆ν2
= 〈N+〉HS + z〈N−〉HS (15)
for the relevant chemical potential ∆ν2 by means of successive approximations. The procedure of
searching for a solution of equations (14)–(15) is described in [20].
As a result, in the first nontrivial approximation corresponding to ∆ν1 = 0 we find the following
expression for ∆ν2 [25]
∆ν2 =
√
1 + z2
2V [M++ + 2zM+− + z2M−−]
∑
k
1
det [1 + ΦCM2]
×
(
βφ̃C
++(k)S1 + βφ̃C
−−(k)S2 + 2βφ̃C
+−(k)S3
)
, (16)
where
S1 = M+++ + zM++− , S2 = M+−− + zM−−− , S3 = M++− + zM+−− . (17)
Apart from Mα1α2
formulas (16)–(17) include the third order cumulants Mα1α2α3
or equivalently
the third order connected correlation functions of the RS. In the case of a two-component hard-
sphere system, the analytical expressions for a second order cumulant can be obtained in the
Percus-Yevick approximation using the Lebowitz’ solution [31, 32]. The corresponding formulas
for Mα1α2
(k = 0) can be found in [25]. In order to derive the expressions for the third order
cumulants one can use the recurrent relation
Mα1α2...αn
= Mα1α2...αn
(0, . . .) =
∂Mα1α2...αn−1
(0, . . .)
∂ν0
αn
, (18)
where ν0
αi
is the MF value of chemical potential ναi
that due to the electroneutrality condition
coincides with the hard-sphere chemical potential of the αith species.
Finally, the expression for the full chemical potential ν2 can be written as follows
ν2 = νHS
2 + νS
2 + ∆ν2 , (19)
where
νHS
2 =
νHS
+ + zνHS
−√
1 + z2
(20)
23004-5
O.V. Patsahan, T.M. Patsahan
with νHS
+ (νHS
− ) being the hard-sphere chemical potential of the αth species and νS
2 being the
combination of the self-energy parts of chemical potentials ν+ and ν−
νS
2 = − 1
2V
√
1 + z2
∑
k
(
βφ̃C
++(k) + zβφ̃C
−−(k)
)
. (21)
The explicit expressions for νHS
2 and νS
2 can be obtained using the results of [32] supplemented by
the electroneutrality condition and are presented in [25].
Below we use formulas (16)–(21) for the study of the gas-liquid phase equilibria in asymmetric
PMs.
3. Results
3.1. Monovalent PMs with size asymmetry
First we consider a monovalent PM with size asymmetry corresponding to z = 1 and λ 6= 1.
Because of symmetry with respect to the exchange of + and − ions, only λ < 1 (or λ > 1) need
be considered in this case.
Figure 1. Coexistence curves of the monovalent PM at different values of λ.
Based on the expressions (16)–(21) (at z = 1) supplemented by the Maxwell construction we
calculate the coexistence data. Figure 1 demonstrates the calculated coexistence curves of the
monovalent size-asymmetric PM at λ = 1.0, 0.75, 0.5, 0.25. As is seen, the region of coexistence
narrows when the size asymmetry increases. This effect becomes increasingly pronounced as λ
becomes smaller that is in qualitative agreement with the simulation findings [7, 8].
Estimates of the critical temperature and the critical density are given by their values for which
the maxima and minima of the van der Waals loops coalesce. The calculated critical parameters
are reported in table 1. Similar to the simulation results, both the critical temperature T ∗
c and the
critical total number density ρ∗
c decrease when the size asymmetry increases.
As was found from simulations [13], the critical volume fraction ηc of a monovalent PM demon-
strates an interesting nonmonotonic behaviour: when the size asymmetry increases, the volume
fraction is seen to increase before it begins to rapidly decrease. For an arbitrary z, the volume
fraction defined as
η =
π
6
ρ
(1 + z)
(σ3
+ + zσ3
−)
23004-6
Phase equilibria of asymmetric primitive models
Table 1. Critical parameters T ∗
c = kBTσ±/q2 and ρ∗
c = ρcσ
3
± of the monovalent PM for different
values of λ.
λ T ∗
c 102ρ∗c
1.0 0.0848 0.907
0.75 0.0831 0.816
0.5 0.0786 0.637
0.25 0.0709 0.433
0.2 0.0682 0.370
0.15 0.0644 0.291
0.1 0.0586 0.195
is related to the reduced total number density ρ∗ (see (11)) by the relation
η =
4π
3
ρ∗
(z + λ3)
(1 + z)(1 + λ)3
.
Our results for the dependence of ηc on size asymmetry at z = 1 are shown in figure 2.
η
λ
Figure 2. The critical volume fraction of the monovalent PM as a function of size asymmetry.
The appearance of the first maximum (at λ = 1) is due to the fact that the expression for
∆ν2 given by (16) tends to the random phase approximation when λ → 1. As was shown in [25],
equation (16) reduces to the RPA in the limiting case of the RPM (z = 1 and λ = 1). For λ > 2,
the calculated trend of ηc is similar to that found in simulations. Moreover, our results indicate
that ηc changes its trend at λ∗ ≈ 4 while the simulations yield λ∗ ≈ 4.26 (|δ∗| ≈ 0.62).
3.2. Size- and charge-asymmetric PMs
Now we use formulas (16)–(21) for the study of the size- and charge-asymmetric PMs with
z 6= 1 and λ 6= 1. In particular, we consider 2:1 and 3:1 models. As before, in order to calculate the
coexistence curves and the corresponding critical parameters we apply the Maxwell construction.
The phase coexistence envelopes for 2:1 systems with size asymmetries are presented in figure 3
for λ < 1 and in figure 4 for λ > 1. Similar to the monovalent case, the coexistence regions become
23004-7
O.V. Patsahan, T.M. Patsahan
Figure 3. Coexistence curves of the 2:1 PM at
different values of λ (λ 6 1).
Figure 4. Coexistence curves of the 2:1 PM at
different values of λ (λ > 1).
Figure 5. Coexistence curves of the 3:1 PM at
different values of λ (λ 6 1).
Figure 6. Coexistence curves of the 3:1 PM at
different values of λ (λ > 1).
Table 2. Critical parameters T ∗
c and ρ∗
c of the 2 : 1 PM for different values of λ.
λ T ∗
c 102ρ∗c
0.2 0.054 0.319
0.33 0.0587 0.458
0.5 0.0614 0.545
0.67 0.0630 0.619
1 0.0640 0.720
1.5 0.0611 0.616
2 0.0583 0.528
3 0.0553 0.461
5 0.0529 0.436
narrower with the increase of size asymmetry. The corresponding critical parameters decrease when
size asymmetry increases that qualitatively agrees with simulation results [10]. The calculated data
for critical temperatures and critical densities are given in table 2. As far as we know, no simulation
results are available regarding the effect of size asymmetry on the coexistence region for z:1 PMs.
23004-8
Phase equilibria of asymmetric primitive models
Figures 5 and 6 demonstrate the effects of size asymmetry on the phase diagrams of 3:1 systems.
The estimated values of the critical parameters are presented in table 3. Again, we can observe
both the decrease of the critical parameters and the reduction of the coexistence regions with the
increase of size asymmetry. As is seen from figure 5, our results demonstrate the phase transition
at λ = 0.2. This contradicts the prediction made in [11] on the basis of an extrapolation of the
simulation data. As was suggested in [11], the phase transition vanishes at λ ' 0.2 (or equivalently
at δ ' −0.67) in the case of 3:1 PM. Thus, this issue deserves further investigation.
Table 3. Critical parameters T ∗
c and ρ∗
c of the 3:1 PM for different values of λ.
λ T ∗
c 102ρ∗c
0.2 0.0441 0.380
1.0 0.0470 0.549
1.5 0.0443 0.465
2.0 0.0420 0.392
3.0 0.0395 0.335
4. Summary
We have studied the effects of size and charge asymmetry on the gas-liquid phase diagrams
of two-component PMs. For this purpose we have used the CVs based theory that enables us to
derive an exact expression for the functional of grand partition function of the model and on this
basis to develop the perturbation theory. As was shown in [20], the well-known approximations for
the free energy, in particular Debye-Hückel limiting law and MSA, can be reproduced within the
framework of this theory.
In this paper, extending our previous studies [25], we have calculated the coexistence curves
for the 1:1, 2:1 and 3:1 PMs at different values of the size asymmetry parameter. In all cases
considered our results have demonstrated the coexistence region narrowing with the increase of
size asymmetry. Since such a behaviour qualitatively agrees with the simulation results for the
monovalent PMs we expect that our predictions for the 2:1 and 3:1 PMs are also qualitatively
correct. Moreover, we have calculated the critical temperatures and critical densities for the 3:1
PM at λ = 0.2, 1, 1.5, 2, 3. As before, our theory in this case predicts a decrease of the both critical
parameters with the increase of size asymmetry.
Summarizing, we can state that the theory developed for the determination of the relevant
chemical potential has for the first time enabled us to get a qualitatively correct description of the
effect of size asymmetry on the phase diagrams of asymmetric PMs without directly including the
ionic association.
Acknowledgement
It is a great pleasure to thank Taras Bryk for giving us the opportunity to contribute to the
Festschrift dedicated to the 50th birthday of Ihor Mryglod.
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Фазова рiвновага примiтивних моделей iонних плинiв iз
асиметрiєю в розмiрах i зарядах: метод колективних
змiнних
О.В. Пацаган, Т.М. Пацаган
Iнститут фiзики конденсованих систем НАН України, вул. Свєнцiцького, 1, 79011 м. Львiв
Ми вивчаємо фазовi дiаграми z:1 примiтивних моделей з асиметрiєю в розмiрах, використовуючи
теорiю, що ґрунтується на методi колективних змiнних. Використовуючи явний вираз для хiмiчного
потенцiалу, спряженого до параметра порядку, який враховує кореляцiйнi ефекти до третього по-
рядку включно, ми розглядаємо декiлька часткових випадкiв асиметричних примiтивних моделей
(системи 1:1, 2:1 та 3:1). Обчислено кривi спiвiснування та критичнi параметри як функцiї розмiр-
ного коефiцiєнту λ = σ+/σ−. Наша теорiя, подiбно до комп’ютерного моделювання, передбачає
звуження областей спiвiснування i зменшення критичних параметрiв T ∗
c i ρ∗c з ростом асиметрiї в
розмiрах.
Ключовi слова: примiтивнi моделi iз асиметрiєю в розмiрах i зарядах, кривi спiвiснування,
критична точка газ-рiдина, метод колективних змiнних
23004-10
Introduction
Theoretical background
Model
Functional integral. The Gaussian approximation
Beyond the Gaussian approximation
Results
Monovalent PMs with size asymmetry
Size- and charge-asymmetric PMs
Summary
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