Crossover behavior in fluids with Coulomb interactions
Crossover behavior in fluids with Coulomb interactions
O.V. Patsahan,1 J.-M. Caillol,2 and I.M. Mryglod1
1Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine,
1 Svientsitskii Str., 79011 Lviv, Ukraine
2Laboratoire de Physique Théorique CNRS UMR 8627,
Bât. 210 Université de Paris-Sud 91405 Orsay Cedex, France
(Dated: October 30, 2018)
Abstract
According to extensive experimental findings, the Ginzburg temperature tG for ionic fluids differs
substantially from that of nonionic fluids [Schröer W., Weigärtner H. 2004 Pure Appl. Chem. 76
19]. A theoretical investigation of this outcome is proposed here by a mean field analysis of the
interplay of short and long range interactions on the value of tG. We consider a quite general
continuous charge-asymmetric model made of charged hard spheres with additional short-range
interactions (without electrostatic interactions the model belongs to the same universality class
as the 3D Ising model). The effective Landau-Ginzburg Hamiltonian of the full system near its
gas-liquid critical point is derived from which the Ginzburg temperature is calculated as a function
of the ionicity. The results obtained in this way for tG are in good qualitative and sufficient
quantitative agreement with available experimental data.
http://arxiv.org/abs/0704.0544v1
I. INTRODUCTION
It is known that electrostatic forces determine the properties of various systems: phys-
ical as well as chemical or biological. In particular, the Coulomb interactions are of great
importance when dealing with ionic fluids i.e., fluids consisting of dissociated cations and
ions. In most cases the Coulomb interaction is the dominant interaction and due to its
long-range character can substantially affect the critical properties and the phase behavior
of ionic systems. Thus, the investigations concerning these issues are of great fundamental
interest and practical importance.
Over the last ten years, both the phase diagrams and the critical behavior of ionic solu-
tions have been intensively studied using both experimental and theoretical methods. These
studies were stimulated by controversial experimental results, demonstrating the three types
of the critical behavior in electrolytes solutions: (i) classical (or mean-field) and (ii) Ising-like
behavior as well as (iii) crossover between the two [1, 2, 3, 4, 5, 6]. In accordance with these
peculiarities, ionic solutions were conventionally divided into two classes, namely: “solvo-
phobic” systems with Ising-like critical behavior in which Coulomb forces are not supposed
to play a major role (the solvent is generally characterized by high dielectric constant) and
“Coulombic” systems in which the phase separation is primarily driven by Coulomb in-
teractions (the solvent is characterized by low dielectric constant). Hence the criticality
of the Coulombic systems became a challenge for theory and experiment. A theoretical
model which demonstrates the phase separation driven exclusively by Coulombic forces is
a restricted primitive model (RPM) [7, 8]. In this model the ionic fluid is described as
an electroneutral binary mixture of charged hard spheres of equal diameter immersed in a
structureless dielectric continuum. Early studies [9, 10, 11] established that the model has a
gas-liquid phase transition. A reasonable theoretical description of the critical point in the
RPM was accomplished at a mean-field (MF) level using integral equation methods [8, 12]
and Debye-Hückel theory [13]. Due to controversial experimental findings, the critical be-
havior of the RPM has been under active debates [14, 15, 16, 17, 18, 19, 20, 21, 22, 23] and
strong evidence for an Ising universal class has been found by recent simulations [21, 24, 25]
and theoretical [26, 27, 28, 29] studies.
In spite of significant progress in this field, the criticality of ionic systems are far from
being completely understood. The investigation of more complex models is very important
in understanding the nature of critical behavior of real ionic fluids demonstrating both the
charge and size asymmetry as well as other complexities such as short-range attraction. A
description of a crossover region when the critical point is approached is of particular interest
for such models. Based on the experimental findings one can suggest that in ionic fluids
the temperature interval of crossover regime, characterized by the Ginzburg temperature,
is much smaller than observed in nonionic systems [5]. In particular, a sharp crossover was
reported for the systems Na−NH3 [30] (see also [31, 32, 33]). The analysis of experimental
data for various ionic solutions confirmed that such systems generally exhibit crossover
or, at least a tendency to crossover from the Ising behavior asymptotically close to the
critical point, to the mean-field behavior upon increasing distance from the critical point
[34]. Moreover, the systematic experimental investigations of the ionic systems such as tetra-
n-butylammonium picrate, Bu4NPic, (for tetra-n-butylammonium picrate we will follow the
notations from [5, 6]) in long chain n-alkanols with dielectric constant ranging from 3.6 for
1-tetradecanol to 16.8 for 2-propanol suggest an increasing tendency for crossover to the
mean-field behavior when the Coulomb contribution becomes essential [5, 6, 35]. They also
indicate that the ”Coulomb limit” reduced temperature of the RPM Tc ≃ 0.05 is valid for
the almost non-polar long chain alkanols [6, 35]. It has been stressed [35] that for solutions
of Bu4NPic in 1-alkanols, the upper critical solution points are found to increase linearly
with the chain length of the alcohols (that corresponds to the decrease of dielectric constant
of the solvent). The experimental data for the critical points and the dielectric permittivities
for solutions of Bu4NPic in 1-alkanols are given in Table 1 [35].
Theoretically the crossover behavior in ionic systems was firstly studied for the RPM
[14, 15, 16]. The results obtained for the Ginzburg temperature were similar to those found
for simple fluids in comparable fashion that is in variance to what is expected from the ex-
periments [5, 6]. Nearly at the same time in [36] the crossover behavior of the lattice version
of a fluid exhibiting the Ising behavior was studied as additional symmetrical electrostatic
interactions were turned on. Based on the microscopic ground, the effective Hamiltonian
in terms of the fluctuating field conjugate to the number density was derived in this work.
Then, the crossover between the mean-field and Ising-like behavior was estimated using the
Ginzburg criterion. The resulting crossover temperature calculated as function of the ionicity
I, which defines the strength of the Coulomb interaction relative to the short-range inter-
action, indicates its weak dependence but with the trends correlating with those observed
TABLE I: The experimental parameters of the critical points (critical temperature Tc, critical mass
fraction wc) and the corresponding dielectric constants ǫ for solutions of Bu4NPic in 1-alkanols [35].
Solvent ǫ(Tc) Tc/K wc
1-oktanol 9.5 298.55 0.336
1-nonanol 7.9 308.64 0.325
1-decanol 6.4 318.29 0.3152
1-undecanol 5.4 326.98 0.303
1-dodecanol 4.7 335.91 0.2951
1-tridecanol 4.3 342.35 0.284
1-tetradecanol 3.6 351.09 0.2721
experimentally.
In this paper we are also interested in the critical behavior of ionic fluids. In particular, we
study the effect of the interplay of short-range and long-range interactions on the crossover
behavior in such systems. We consider a continuous version of the charge-asymmetric ionic
fluid in which both the long-range Coulomb and short-range van-der Waals-like interactions
are included. Following [36] we introduce the ionicity
I = 1
|q1q2|
kBTǫσ
, (1)
where qi is the charge on ion i, kB is the Boltzmann constant, T is the temperature, σ is
collision diameter and ǫ is the dielectric constant. Then we derive the effective Hamiltonian
of the charge-asymmetric model in the vicinity of the gas-liquid critical point. As in [36],
the coefficients obtained for the effective Hamiltoninan have the forms of expansions in the
ionicity but with new terms that appear in this case. Based on this Hamiltonian we estimate
the Ginzburg temperatures as functions of the ionicity.
The layout of the paper is as follows. In Section 2 we introduce a continuous charge-
asymmetric model with additional short-range attractive interactions included. We derive
here the functional representation of the grand partition function of the model in terms of the
fluctuating fields ϕSk and ϕ
k conjugate to the total density and charge density, respectively.
Section 3 is devoted to the derivation of the effective GLW Hamiltonian in the vicinity of
the critical point. In Section 4 we calculate the Ginzburg temperature as a function of the
ionicity for different values of the range of the attractive potential. We conclude in Section 5.
II. BACKGROUND
A. Model
Let us start with a general case of a classical charge-asymmetric two-component system
consisting of N particles among which there are N1 particles of species 1 and N2 particles
of species 2. The pair interaction potential is assumed to be of the following form:
Uαβ(r) = φ
αβ (r) + φ
αβ (r) + φ
αβ(r), (2)
where φHSαβ (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. Thermody-
namic and structural properties of the reference system are assumed to be known. φSRαβ (r)
is the potential of the short-range (van-der-Waals-like ) attraction. φCαβ(r) is the Coulomb
potential: φCαβ(r) = qαqβφ
C(r)/ǫ, where φC(r) = 1/r and ǫ is the dielectric constant. The
solution is made of both positive and negative ions so that the electroneutrality condition
is satisfied,i.e.
α=1 qαcα = 0, where cα is the concentration of the species α, cα = Nα/N .
The ions of the species α = 1 are characterized by their hard sphere diameter σ1 and their
electrostatic charge +q0 and those of species α = 2, characterized by diameter σ2, bear
opposite charge −zq0 (q0 is elementary charge and z is the parameter of charge asymmetry).
In general, the two-component system of hard spheres interacting via the potential φSRαβ (r)
can exhibit both the gas-liquid and demixion critical points which belong to the 3D Ising
model universal class.
We consider the grand partition function (GPF) of the system which can be written as
follows:
Ξ[να] =
α=1,2
exp(ναNα)
(dΓ) exp
Uαβ(rij)
. (3)
Here the following notations are used: να is the dimensionless chemical potential, να =
βµα−3 lnΛ, µα is the chemical potential of the αth species, β is the reciprocal temperature,
Λ−1 = (2πmαβ
−1/h2)1/2 is the inverse de Broglie thermal wavelength; (dΓ) is the element
of configurational space of the particles: (dΓ) =
α dΓα, dΓα = dr
2 . . .dr
Let us introduce the operators ρ̂Sk and ρ̂
ρ̂Sk =
ρ̂k,α ρ̂
qαρ̂k,α,
which are combinations of the Fourier transforms of the microscopic number density of the
species α: ρ̂k,α =
i exp(−ikrαi ). In this case the part of the Boltzmann factor entering eq.
(3) which does not include hard sphere interactions can be presented as follows:
(Uαβ(rij)− φHSαβ (rij))
= exp
(φ̃SS(k)ρ̂
+φ̃DD(k)ρ̂
−k + 2φ̃SD(k)ρ̂
−k) +
(φ̃SRαα (k) + q
C(k))
, (4)
where
φ̃SS(k) =
(1 + z)2
z2φ̃SR11 (k) + 2zφ̃
12 (k) + φ̃
22 (k)
φ̃DD(k) =
(1 + z)2
φ̃SR11 (k)− 2φ̃SR12 (k) + φ̃SR22 (k)
+ φ̃C(k)
φ̃SD(k) =
(1 + z)2
zφ̃SR11 (k) + (1− z)φ̃SR12 (k)− φ̃SR22 (k)
with φ̃X...αβ (k) being a Fourier transform of the corresponding interaction potential defined by
φ̃X...αβ (k) =
drφX...αβ (r) exp(−ikr), φX...αβ (r) =
φ̃X...αβ (k) exp(ikr).
Now we simplify our model assuming that
• The hard spheres will all be of the same diameter σα = σ.
• φ̃SR++(k) = φ̃SR−−(k) = φ̃SR+−(k) = φ̃SR(k).
With these restrictions the uncharged system can only exhibit a gas-liquid critical point and
a possible demixion is ruled out.
Taking into account the assumptions mentioned above we thus have
φ̃SS(k) = φ̃
SR(k) < 0, φ̃DD(k) = φ̃
C(k) > 0, φ̃SD(k) ≡ 0.
Finally it will be convenient to introduce the effective range bSR of short-range interactions
through the relations
φ̃SR(k) = φ̃SR(0)
1− (bSR k)2
+O(k4) . (6)
B. Functional representation of the grand partition function of an ionic model
Let us take advantage of the properties of Gaussian functional integrals to rewrite
w̃S(k)ρ̂
(dϕS) exp
[w̃S(k)]
ρ̂Skϕ
w̃C(k)ρ̂
(dϕD) exp
[w̃C(k)]
ϕDk ϕ
ρ̂Dk ϕ
NwS =
(dϕS) exp
[w̃S(k)]
NwC =
(dϕD) exp
[w̃C(k)]
ϕDk ϕ
(dϕA) =
dϕAk =
d(ℜϕAk )d(ℑϕAk ), A = S,D.
In the above equations we also introduced the notations w̃S(k) = −βφ̃SS(k)/V and
w̃C(k) = φ̃
C(k)/V .
As a result, we can rewrite Ξ[να] in the form of a functional integral
Ξ[να] =
(dϕS)(dϕD) exp
−H[να, ϕS, ϕD]
, (7)
where the action H reads as
H[να, ϕS, ϕD] =
[w̃S(k)]
[w̃C(k)]
ϕDk ϕ
− ln ΞHS[νS + ϕS, νD + iβ1/2ϕD], (8)
1 + z
ν̄1 +
1 + z
ν̄2, ν̄D =
q0(1 + z)
(ν̄1 − ν̄2). (9)
where the ”renormalized” chemical potentials να are defined as
να = να +
−w̃S(k) + βq2αw̃C(k)
, α = 1, 2. (10)
Let us define ∆νS = νS−ϕS0 and ϕ̃Sk = ∆νS+ϕSk with ϕS0 chosen as the chemical potential
of the hard spheres. This leads to the relation
νS + ϕ
S = ϕS0 + ϕ̃
S. (11)
Now we present ln ΞHS[. . .] in the form of a cumulant expansion
ln ΞHS[. . .] =
k1,...,kn
0 , νD; k1, . . . , kn]ϕ̃
. . . ϕ̃Dkin
ϕ̃Skin+1
. . . ϕ̃Sknδk1+...+kn , (12)
where M
0 , νD; k1, . . . , kn] is the nth cumulant (or the nth order truncated correlation
function) defined by
0 , νD; k1, . . . , kn] =
∂n ln ΞHS[. . .]
. . . ∂ϕ̃D
kin+1
. . . ∂ϕ̃S
. (13)
In particular it follows from (13) that
0 = lnΞHS[ϕ
0 , νD]. (14)
The expressions for the cumulants of higher order (for in ≤ 4) are given in Appendix A.
It should be noted that, contrary to [36], (12) includes all powers (even and odd) of the
field ϕSk conjugate to the total number density. It should be clear that the coefficients in the
cumulant expansion (12) depend on the chemical potential (or, equivalently, on the density).
III. EFFECTIVE HAMILTONIAN IN THE VICINITY OF THE CRITICAL
POINT
Taking into account (12) we can rewrite (7)-(8) as follows
Ξ[να] =
(dϕ̃S) exp
[w̃S(k)]
ϕ̃Skϕ̃
+ [w̃S(0)]
∆νSϕ̃S0 +
k1,...,kn
0 , νD]ϕ̃
. . . ϕ̃Skn
×δk1+...+kn)V[ϕ̃Sk], (15)
where
H = 1
[w̃S(0)]
(∆νS)2 − ln ΞHS[ϕS0 ],
V[ϕ̃Sk] =
(dϕ̃D) exp
[w̃C(k)]
ϕ̃Dk ϕ̃
k1,k2,k3
ϕ̃Dk2ϕ̃
δk1+k2+k3 +
k1,...,k4
ϕ̃Dk2ϕ̃
×ϕ̃Sk4δk1+...+k4 +
k1,...,k4
ϕ̃Dk2ϕ̃
ϕ̃Sk4δk1+...+k4 + . . .
. (16)
It is worth noting here that unlike to the case considered in [36] we obtain in (16) terms
proportional to
and ϕ̃S
. While the former is connected with an absence
of a lattice symmetry, the the latter stems from charge asymmetry.
Our aim now is to derive the effective Landau-Ginzburg (LG) Hamiltonian. Since we are
interested in the gas-liquid critical point, this Hamiltonian should be written in terms of
fields ϕ̃Sk conjugate to the fluctuation modes of the total number density.
To this end we integrate out ϕ̃Dk in (16) using a Gaussian measure. As a result, we can
present V[ϕ̃Sk] as follows:
V[δϕ̃Sk] =
1 + 〈A〉G +
〈A2〉G +
〈A3〉G + . . .
, (17)
where 〈. . .〉G means
〈. . .〉G =
(dϕD) . . . exp
W̃C(k)ϕ
with W̃C(k) given by
W̃C(k) = [w̃C(k)]
+ y2G̃1 (18)
and y2 being the ionicity introduced by (1): y2 = I
Taking into account (1) and the recurrence formulas of Appendix A A may be written
as a formal expansion in terms of y2
A = −y
k1,k2,k3
G̃2(k1, k2 + k3)ϕ
ϕDk2ϕ
δk1+k2+k3 −
k1,...,k4
G̃3(k1, k2, k3 + k4)
×ϕDk1ϕ
ϕSk3ϕ
δk1+...+k4 −
(1− z)√
k1,...,k4
G̃2(k1, k2 + k3 + k4)
×ϕDk1ϕ
ϕDk3ϕ
δk1+...+k4 + . . . . (19)
In (18)-(19) the “tilde” over ϕ
was omitted for the sake of simplicity.
It should be mentioned that the dependence of G̃n(k1, k2, . . . , kn) on the ki is very compli-
cated. Since we consider here the behavior of the system near the critical point the limiting
case of ki = 0 is of particular interest. Therefore, we substitute in (17)
G̃n(k1, k2, . . . , kn) ≡ G̃n(0, . . .) n ≥ 3
G̃2(k) = G̃2(0)(1 + g
2k2), (20)
G̃22(0)
2G̃2(0)
, G̃22(0) =
∂2G̃2(k)
|k=0. (21)
Having integrated out eq. (17) Ξ[να] takes the form:
Ξ[να] =
1 + y2〈N〉HSw̃C(k)
(dϕS) exp
−Heff [ϕS]
Heff [ϕS] = −
k1,...,kn
. . . ϕ̃Sknδk1+...+kn, (22)
where we have for the coefficients an
a0 = −H, (23)
a1 = 〈N〉HS + [w̃S(0)]−1∆νS −
G̃2(0)
∆̃(q) +
3G̃3(0) +
(1− z)2 − 2z
×G̃2(0)
∆̃(q)
, (24)
a2 = −[w̃S(k)]−1 + G̃2(k)−
G̃3(0)
∆̃(q) +
[G̃2(0)]
∆̃(q)∆̃(| k + q |), (25)
a3 = G̃3(0)−
G̃4(0)
∆̃(q) +
y4G̃2(0)G̃3(0)
∆̃2(q), (26)
a4 = G̃4(0)−
G̃5(0)
∆̃(q) +
3[G̃3(0)]
2 + 4G̃2(0)G̃4(0)
∆̃2(q), (27)
and the propagator ∆̃(q) is written as
∆̃(q) = ∆̃(q; y2) = [W̃C(q)]
w̃C(q)
1 + y2〈N〉HSw̃C(q)
. (28)
Coefficients (23)-(27) have the form of a formal expansion in terms of the ionicity I = y2.
In our study all terms which do not exceed the fourth order of y are kept. The ionicity is
small enough for large values of the dielectric constant and increases with its decrease. From
this point of view we can consider the expansions in (23)-(27) for large values of y2 only as
formal ones. It should be also noted that ∆̃(q) ∼ 1/y2 (see (28)) for large values of y2.
Let us introduce
rSR = [G̃2(0)w̃S(0)]
−1 − 1 = T − Tc,0
, (29)
where Tc,0 = Tc(I = 0) is the mean-field critical temperature of the uncharged system.
Taking into account (29) we can rewrite −Heff as follows:
−Heff [ϕS] = −1
[w̃S(0)]
(∆νS)2 + lnΞHS[ϕ
r0 + τ
k1,k2,k3
ϕSk1ϕ
ϕSk3δk1+k2+k3 −
k1,k2,k3,k4
ϕSk1ϕ
ϕSk3ϕ
×δk1+k2+k3+k4 − h0ϕS0 , (30)
where ϕSk =
G̃2(0)ϕ
k and the following notations were introduced:
r0 = rSR +
G̃3(0)
G̃2(0)
∆̃(q)− y
G̃2(0)
∆̃2(q) (31)
τ 20 = τ
G̃2(0)
∆̃(q)∆̃(2)(q) (32)
v0 = −
[G̃2]1/2
G̃3(0)
G̃2(0)
G̃4(0)
G̃2(0)
∆̃(q) +
G̃3(0)
∆̃2(q)
, (33)
u0 = −
G̃4(0)
G̃2(0)
G̃5(0)
G̃2(0)
∆̃(q) +
∆̃2(q)
G̃2(0)
3[G̃3(0)]
2 + 4G̃2(0)G̃4(0)
, (34)
h0 = −[G̃2(0)]1/2
〈N〉HS + [w̃S(0)]−1∆νS
G̃2(0)
∆̃(q) +
G̃3(0)
G̃2(0)
(1− z)2 − 2z
∆̃(q)
 , (35)
where
− τ 2SR = g2 +
b̄2SR
G̃2(0) [w̃S(0)]
with b̄2SR = b
SRw̃S(0) and ∆̃
(2)(q) = ∂2∆̃(| k+ q |)/∂k2|k=0.
Finally, we present (30) as follows:
−Heff [ϕS] = −1
[w̃S(0)]
(∆νS)2 + lnΞHS[ϕ
r + τ 2k2
〈N〉1/2HS
k1,k2,k3
ϕSk1ϕ
ϕSk3δk1+k2+k3 −
〈N〉HS
k1,...,k4
ϕSk1ϕ
ϕSk3ϕ
×δk1+k2+k3+k4 − h〈N〉
0 , (37)
r = r0, τ
2 = τ 20 , v =
〈N〉1/2HS u =
〈N〉HS, h = h0〈N〉−1/2HS .
At the critical point the following equalities hold
r = 0, v = 0, h = 0,
which give the equations for the critical parameters i.e., the temperature, the density and
the chemical potential at the critical point.
Equation (37) gives the effective GLW Hamiltonian of the system (2) in the vicinity of
the critical point. We are now in position to extract from eq. (37) the Ginzburg temperature
as a function of the ionicity.
Now let us specify the short-range attraction, φSR(r), in the form of the square-well
potential
φSR(r) =
0, 0 ≤ r < σ
−ε, σ ≤ r < λσ
0, r ≥ λσ
It is worth noting here that the system of hard spheres interacting through the potential
φSR(r) with λ = 1.4− 1.7 reasonably models most simple fluids [37]. The Fourier transform
of φSR(r) for the case of the Weeks-Chandler-Andersen (WCA) regularization inside the
hard core [38] has the form:
φ̃SR(k) = φ̃SR(0)
(λx)3
[−λx cos(λx) + sin(λx)], (38)
where x = kσ and φ̃SR(0) = −εσ3 4π
To be consistent we also use the WCA regularization scheme for the Coulomb potential
which yields
φ̃C(x) = 4π sin(x)/x3. (39)
IV. GINZBURG TEMPERATURE
Following [36] we can present the Ginzburg temperature by
tG[ηc(y), y] ≃
u2(y)
[1 + t0(y)]τ 6(y)
, (40)
but in our case all quantities u, t0 and τ
2 should be estimated at critical density ηc(y):
u(y) = u(ηc(y), y), t0(y) = t0(ηc(y), y), τ
2(y) = τ 2(ηc(y), y).
The density η enters the expressions for u, t0 and τ
2 through the structure factors S̃n. A
well-known criterium by Ginzburg predicts that the mean-field theory is valid only when
tG <<| t | where t = T−Tc(y)Tc(y) and Tc(y) are the mean-field reduced temperature and the
mean-field critical temperature of the charged system at η = ηc(y), respectively.
In (40) t0(y) measures the increase of the mean-field temperature of the charged system
in respect to the uncharged system
t0(y) =
Tc(y)
− 1 (41)
which, for the model under consideration has the form:
t0(y) = −
G̃3(0)
G̃2(0)
∆̃(q) +
G̃2(0)
∆̃2(q). (42)
Taking into account that [
G̃2(0)w̃C(0)
and equation (41), we can rewrite (36) as follows:
− τ 2SR = g2 +
w̃C(0)
(1 + t0(y)). (43)
For the uncharged model the Ginzburg temperature reduces to
tG(I = 0)) =
S̃42 [τSR(I = 0)]6
where S̃n is given by (45) and τSR(I = 0) = τSR(t0 = 0).
First we calculate the critical density from the equation v = 0. To this end we take into
account (33), (39) and the formulas of Appendix D. As a result, we obtain the dependence
of the dimensionless critical density ηc (η = πρσ
3/6) on the ionicity I = y2 (see Fig. 1).
0 4 8 12 16 20
0.105
0.110
0.115
0.120
0.125
0.130
FIG. 1: Critical density as a function of I.
In order to calculate the chemical potential at the critical point we introduce ∆ν =
∆νS − νSMF , where
νSMF = −〈N〉HSw̃S(0)
is the mean-field value of the chemical potential νS(0). ∆νc is obtained from the condition
h = 0; taking into account (35) it yields :
∆νc =
∆̃(q)− y
3S̃3 +
(1− z)2 − 2z
∆̃(q)
, (44)
where
S̃n(ηc; 0) = G̃n/〈N〉HS (45)
is the nth particle structure factor at the critical density ηc(I) when ki = 0. In Fig. 2 ∆νc
is displayed as a function of the ionicity for different values of the parameter z.
Now we calculate τ 2, u, t0 and tG at η = ηc using (32), (34), (38)-(39), (42) and formulas
from Appendices B-D.
The dependence of τ 2 on I at different values of the parameter λ is plotted in Fig. 3.
The explicit formula for τ 2SR is given in Appendix C. The coefficient u and the shift in the
mean-field critical temperature, t0, as functions of I are plotted in Figs. 4 and 5. As is seen,
quantities τ 2, t0 and u are increasing functions of I in the whole region under consideration
and their dependencies of I are at variance with those obtained in [36] for the lattice model.
0 5 10 15 20
0.0 0.2 0.4 0.6 0.8 1.0
FIG. 2: ∆νc as a function of I calculated from (44) for different values of z (η = ηc). The inset
depicts the behavior of ∆νc close to the origin.
0 5 10 15 20
 λ=1.3
 λ=1.5
 λ=1.7
 λ=2.0
 λ=2.2
 λ=2.5
FIG. 3: The dependence of τ2 on the ionicity for different λ (η = ηc).
Despite this fact, the behavior of the Ginzburg temperature as a function of I calculated
in this work is qualitatively similar to that found in [36] (see Figs. 6-8). Moreover, as in
[36], the behavior of tG(I) becomes nonmonotonic starting with some value of the attraction
potential range (λ in our case). One can see in Fig. 7 that, for λ = 2, tG first drop off (at
very small values of the ionicity) then increases slightly and at I ≃ 1.23 again starts to
0 5 10 15 20
FIG. 4: The dependence of u on the ionicity I (η = ηc).
0 4 8 12 16 20
FIG. 5: The reduced shift of the mean-field critical temperature, t0, as a function of I at η = ηc.
decrease. In Fig. 8 the ratio of reduced Ginzburg temperatures, tG(I)/tG(0), is shown at
different values of λ. It is worth noting that the non-monotonic behavior of tG(I) becomes
more pronounced as λ increases.
In Table 2 we compare our results for the ionicity dependence of the Ginzburg temper-
ature (at λ = 1.5) with the results obtained in [36] for the lattice model as well as with
experimental data for the crossover temperatures t× (data for I and t× are taken from
[36]). The systems (b)-(d) correspond to the same ionic species Bu4NPic within solvents of
different dielectric constant. As is seen, in this case our results are in good agreement (qual-
0 5 10 15 20
0.00 0.02 0.04 0.06 0.08 0.10
0.0920
0.0925
0.0930
0.0935
0.0940
FIG. 6: The reduced Ginzburg temperature, tG, as a function of I at λ = 1.5 (η = ηc). The inset
depicts the behavior of tG(I) close to the origin.
0 5 10 15 20
0.000
0.002
0.004
0.006
0.008
0.010
0.012
0.00 0.01 0.02 0.03
0.01070
0.01072
0.01074
FIG. 7: The same as in Fig. 6 but at λ = 2.
itative and quantitative) with the experimental findings. The system (d) is Na in NH3 and,
of course, might be described by the potential φSR(r) with the different attraction range λ.
For instance, for λ = 2 we obtain tG(I = 6.97) = 0.8×10−2 (see Fig. 7) that correlates with
the experimental value t× = 0.6× 10−2
0 5 10 15 20
 λ=1.3
 λ=1.5
 λ=1.7
 λ=2.0
 λ=2.2
 λ=2.5
FIG. 8: The ratio tG(I)/tG(0) as a function of the ionicity at different values of λ (η = ηc).
TABLE II: Experimentally assessed crossover temperature, t×, taken from [36]: (a) tetra-n-
butylammonium picrate (Bu4NPic) in 1-tridecanol; (b) Bu4NPic in 1-dodecanol; (c) Bu4NPic in
75% 1-dodecanol plus 25% 1,4-butanediol; (d) Na in NH3; (e) tetra-n-pentylammonium bromide
in water and the reduced Ginzburg temperature, tG, found theoretically in [36] and in this work.
System Ionicity,I t× tG ([36]) tG (this work)
uncharged fluid 0 O(I) 1 ∼ 0.09
(a) 17.9 ∼ 10−3 ∼ 0.712 2.7 × 10−3
(b) 16.8 ∼ 0.9 × 10−2 ∼ 0.717 0.38 × 10−2
(c) 8.9 ∼ 3× 10−2 ∼ 0.777 2.5 × 10−2
(d) 6.97 ∼ 0.6 × 10−2 ∼ 0.807 3.7 × 10−2
(e) ∼ 1.4 O(I) 1 ∼ 0.09
V. SUMMARY
In this paper we study the reduced Ginzburg temperature as a function of the interplay
between the short- and long-range interactions. The ionic fluid is modelled as a charge
asymmetric continuous system that includes additional short-range attractions. The model
without Coulomb interactions exhibits a gas-liquid critical point belonging to the Ising class
of criticality. We derive an effective GLW Hamiltonian for the model whose coefficients have
the form of an expansion in powers of the ionicity. Using these coefficients we calculate
a Ginzburg temperature depending on the ionicity. To this end we introduce a specific
model which consists of charged hard spheres of the same diameter interacting through the
additional square-well potentials. To study the effect of the interplay between short- and
long-range interactions we change, besides the ionicity, the range of the square-well potential.
As a result, we obtain the similar tendency for the reduced Ginzburg temperature as in
[36] when the region of the short-range attraction increases i.e., its nonmonotonic charac-
ter but with different numerical characteristics. However, our results demonstrate a much
faster decrease of the Ginzburg temperature when the ionicity increases. We found a good
qualitative and sufficient quantitative agreement with the experimental findings for Bu4NPic
in n-alkanols. This confirms the experimental observations that an interplay between the
solvophobic and Coulomb interactions alters the temperature region of the crossover regime
i.e., the increase of the ionicity that can be related to the decrease of dielectric constant
leads to the decrease of the crossover region. We suggest that the quantitative discrepancy
of the results for tG obtained in [36] and in this work could be due to the fact, besides
the difference in the symmetry of the two models, that the chemical potential (or density)
dependence of the Hamiltonian coefficients was taken into account explicitly in our case.
It should be noted that in the approximation considered in this paper only the critical
chemical potential depends explicitly on the charge magnitude. In order to obtain the charge
dependence of the other quantities terms of order higher than y2 should be taken into account
into the effective Hamiltonian. Finally, we emphasize that the functional representation (7)-
(8) allows to consider more complicated models in particular models including charge and
size asymmetry.
VI. APPENDICES
A. Recurrence formulas for the cumulants Fourier space.
n (k1, k2, . . . , kn) = G̃n(k1, k2, . . . , kn)
n (k1, k2, . . . , kn) = 0
n (k1, k2, . . . , kn) = (i)
βq2αcαG̃n−1(k1, k2, . . . , |kn−1 + kn|)
n (k1, k2, . . . , kn) = (i)
β3/2q3αcαG̃n−2(k1, k2, . . . , |kn−2 + kn−1 + kn|)
n (k1, k2, . . . , kn) = (i)
q2αcα
G̃n−2(k1, k2, . . . , |kn−2 + kn−1 + kn|)
q4αcα − 3
q2αcα
G̃n−3(k1, k2, . . . , |kn−3 + . . .+ kn|)
where G̃n(k1, k2, . . . , kn) is the Fourier transform of the n-particle truncated correlation
function [39] of a one-component hard sphere system and summation over repeated indices
is meant.
B. The nth-particle structure factors of a one component hard sphere systems in
the Percus-Yevick approximation
S2(0) =
(1− η)4
(1 + 2η)2
, (46)
S3(0) =
(1− η)7(1− 7η − 6η2)
(1 + 2η)5
, (47)
S4(0) =
(1− η)10(1− 30η + 81η2 + 140η3 + 60η4)
(1 + 2η)8
, (48)
S5(0) =
(1− η)13(1− 85η + 957η2 − 1063η3 − 3590η4 − 2940η5 − 840η6)
(1 + 2η)11
C. Explicit expression for τ2SR
Let us write the Ornstein-Zernike equation in the Fourier space
S̃2(k) =
1− ρc̃(k) , (50)
where c̃(k) is the Fourier transform of the Ornstein-Zernike direct correlation function [40]
We have for c̃(k) in the Percus-Yevick approximation [41]
ρc̃(k) = −24η
αk3(sin(k)− k cos(k)) + βk2(2k sin(k)− (k2 − 2) cos(k)− 2)
ηα((4k3 − 24k) sin(k)− (k4 − 12k2 + 24) cos(k) + 24)
/k6, (51)
where
(1 + 2η)2
(1− η)4 , β = −6
η(1 + 1
(1− η)4
From (50) and (51) we get for g2
G̃22(0)
2G̃2(0)
= 0.05η
(4η6 − 27η5 + 84η4 − 146η3 + 144η2 − 75η + 16)
(1 + 2η)2(1− η)4 .
Taking into account (38) we have b2SR/w̃S(0) = 0.1λ
2. As a result, τ 2SR is as follows
τ 2SR = −0.05
(4η6 − 27η5 + 84η4 − 146η3 + 144η2 − 75η + 16)
(1 + 2η)2(1− η)4 + 2λ
2(1 + t0(y))
, (52)
where t0(y) is given by (42).
D. Explicit expressions for the integrals used in equations (31)-(35)
Using
(2π)3
(dk) we can present
∆̃(k) =
dxx2∆(x), (53)
∆̃(k)
π〈N〉HS
, (54)
∆̃(k)∆̃(2)(k) =
32ησ2
π〈N〉HS
dxx∆(x) (2f1(x) + xf2(x)) , (55)
where the following notations are introduced:
∆(x) = sin (x)
x3 + κ∗
sin (x)
, (56)
f1(x) =
x2 (cos (x) x− 3 sin (x))
x6 + 2 κ∗
x3 sin (x) + κ∗
4 − κ∗4 cos2 (x)
f2(x) = −x
x5 sin (x) + x2κ∗
x2 cos2 (x) + 6 cos (x) x4 − 6 κ∗2x sin (x) cos (x)
−12 x3 sin (x) + 6 κ∗2 − 6 κ∗2 cos2 (x)
x9 + 3 κ∗
x6 sin (x) + 3 κ∗
−3 κ∗4x3 cos2 (x) + κ∗6 sin (x)− κ∗6 sin (x) cos2 (x)
with x = kσ and κ∗ = κDσ =
24y2η being the reduced Debye number.
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	Introduction
	Background
	Model
	Functional representation of the grand partition function of an ionic model
	Effective Hamiltonian in the vicinity of the critical point
	Ginzburg temperature
	Summary
	Appendices
	 Recurrence formulas for the cumulants Fourier space.
	The nth-particle structure factors of a one component hard sphere systems in the Percus-Yevick approximation
	Explicit expression for SR2 
	 Explicit expressions for the integrals used in equations (??)-(??)
	References
