Critical behaviour of ionic fluids

14 Pages·0.22 MB·English

by  O. V. Patsahan

Critical behaviour of ionic fluids
O.V. Patsahan
Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine,
1 Svientsitskii Str., 79011 Lviv, Ukraine
E-mail: [email protected]
(Dated: February 1, 2008)
Recentlyweproposedamicroscopicapproachtothedescriptionofthephasebehaviourandcritical
3 phenomenainbinaryfluidmixtures. Itwasbasedonthemethodofcollectivevariables(CV)witha
0 referencesystem. TheapproachallowedustoobtainthefunctionaloftheGinzburg-Landau-Wilson
0
(GLW)Hamiltonian expressedin termsof theCV(fluctuatingdensities). Thecorrespondingset of
2
CV included the variable connected with the order parameter. In this paper we use this approach

n to the study of the critical behaviour of ionic fluids. For the restricted primitive model (RPM) we
a obtainthefunctionalofthegrandpartitionfunctioninthephasespaceofthetwofluctuatingfields
J conjugate to the fluctuating densities. First we calculate the phase diagram of the RPM in the

4 mean-field (MF) approximation and then we do this calculation taking into account the terms of
the higher orders in the effective Hamiltonian. In the both cases the phase diagrams demonstrate

] thegas-liquid(GL)andchargeorderingphaseinstabilities. Inthelattercase,theobtainedvaluefor
t
f theGL criticaltemperatureisingood agreement withtheMCsimulation datawhereasthecritical
o densityisunderestimated. Theexplicitexpression found forthegrandthermodynamicpotentialin
s thevicinity of theGL critical point implies a classical critical behaviour of the RPM.
.
t
a
PACSnumbers: 05.70.Fh,05.70.Jk,02.70.Rr,64.70.Fx
m
-
d
I. INTRODUCTION
n
o
c Nowdays the theory of phase transitions and critical phenomena is well developed in general. It enables one to
[
predict both universal and non-universal properties for many model systems. However, a number of questions still

1 remains open, among which the critical behaviour of ionic fluids is of great interest. Recent experiments have shown
v that both Ising-like and mean-field like criticality can be observed in such systems. For reviews of the experimental
9 and theoretical situation see Refs. [1, 2, 3, 4, 5, 6]. Numerous theoretical and computer simulation studies of the
3 restricted primitive model (RPM), the simplest model for ionic systems, have not provided a clear picture of the
0
thermodynamics in the critical region [7, 8, 9, 10, 11, 12, 13, 14, 15].
1
In this paper we address the issue of the critical behaviour of the RPM using the approach proposed in [16, 17]
0
for the binary symmetrical mixture. The theory has its origin in the approach based on a functional representation
3
0 of a partition function in the collective variables (CV) space [18, 19]. Its particular feature is a choice of the phase
/ space in which the system is considered. Among the independent variables of this space there could be the ones
t
a connected with the order parameters. This phase space is formed by a set of CV. Each of them is a mode of
m density fluctuations corresponding to the specificity of the model under consideration. This approach allows one to
- determine, on microscopic grounds, the explicit form of the effective GLW Hamiltonian and then to integrate the
d
partition function in the neighborhood of the phase transition point taking into account the renormalization group
n
symmetry. As a result, non-classical critical exponents and analytical expressions for thermodynamic functions are
o
obtained [20, 21]. More recently this theory was developed for a binary fluid mixture [16, 17, 22, 23, 24, 25].
c
: The paper consists oftwoparts. Inthe firstpartwe obtainthe functionalofthe grandpartitionfunction (GPF)of
v
i the RPMgiveninthe phasespaceofthe twofluctuating fields: the fluctuating fieldhk=0 conjugatetothe fluctuating
X total number density ρk=0 and the fluctuating field γk conjugate to the fluctuating charge density ck. Restricting
r our considerationto the second powers of γk (taking into account the higher powers of hk=0) we derive the equation
a for the chemical potential. Based on the chemical potential obtained from the linearized equation (that corresponds
to the mean-field (MF) approximation) we calculate the spinodal curve. Its run suggests that two types of phase
instabilities can occur in the RPM. One part of the spinodal is the gas-liquid (GL) type while another one looks
like a λ-line. We obtain the following values of the GL critical point: T∗ = 0.084 and η = 0.005 which agree with
c c
the other MF theories [26]. In order to study the nature of the criticality of the RPM as well as to get the best
estimations for its GL critical point we go beyond the above mentioned approximationtaking into account the terms
of higher orders in the effective Hamiltonian. The second part of the paper is devoted to this end. First, we consider
the Gaussian approximation of the functional of the GPF. It yields the equation for the boundary of stability with
respecttothechargedensityfluctuations. Then,applyingtheprocedureproposedin[16,17]weobtaintheexpression
for the grandthermodynamic potential in the vicinity of the GL critical point as a power series in the field h˜k=0 (up
to h˜4) conjugate to the order parameter. The expression obtained has the form of the Landau free energy. We also
2
calculate the phase diagram demonstrating both GL and charge ordering phase instabilities. The data for the GL
critical point are T∗ =0.0502 and η =0.022.
c c
II. FUNCTIONAL REPRESENTATION OF THE GPF OF THE RPM. AN EQUATION FOR THE
CHEMICAL POTENTIAL; THE PHASE DISGRAM IN THE MF APPROXIMATION
The RPM consists of N = N +N hard spheres of diameter σ with N carrying charges +q and N (= N )
+ − + − +
charges q, in a medium of dielectric constant D. The interaction potential of the RPM has the form
−
if r <σ
Uγδ(r)= ∞qγqδ if r σ , qi =±q.
(cid:26) Dr ≥
We split the potential U (r) into short- and long-range parts using the Weeks-Chandler-Andersen partition [27].
γδ
As a result, we have
if r σ
ψγδ(r)= 0∞ if r >≤σ ,
(cid:26)
qγqδ if r σ
Φγδ(r)= qDγqσδ if r >≤σ .
(cid:26) Dr
This simple form for Φ (r) inside the hard core changes the behaviour of the Fourier transform for large k from
γδ
usual Colombic k−2 to k−3 decay. As was shown [28], this choice of Φ (r) for r < σ produces rapid convergence of
γδ
the seriesofthe perturbationtheoryfor the freeenergy. The FouriertransformofΦ (r)= q2 =Φ (r) hasthe form
γδ Dr C
sinx
βρΦ˜ (x)=24β∗η , (1)
C x3
where β∗ = βq2, β = 1 , η = πρσ3 is fraction density, x=kσ.
Dσ kBT 6
We start with the GPF. We use the formulas obtained within the framework of the CV method for a binary
continuoussystem(see(38)-(42)inAppendixA).InthecaseoftheRPMwehaveV˜(k) 0,U˜(k) 0,W˜(k)=2Φ˜ (k)
C
≡ ≡
and µ =µ (µ is a part of the chemical potential of the ith species, i=+, ). As a result, we can present the
1,+ 1,− 1,i
−
functional of the GPF of the RPM in the form (for details see Appendix A ):
β
Ξ=Ξ0 (dρ)(dc)exp βµ1ρ0 Φ˜C(k)ckc−k J(ρ,c). (2)
Z − V k !
X
Here Ξ is the GPF of the RS. In the case of the RPM, the RS is a one-component hard-sphere system with the
0
diameter σ (potential ψ (r)=ψ(r)).
γδ
ρk and ck are the CV connected with total density fluctuation modes and charge density fluctuation modes,
respectively. µ (µ =(µ +µ )/√2) is determined from the equation
1 1 1,+ 1,−
∂lnΞ N
1
= h i. (3)
∂βµ1 √2
For the RPM J(ρ,c) has the same form as that for the symmetrical binary fluid [17]:
1
J(ρ,c) = Z (dγ)(dω)exp(i2π k (ωkρk+γkck)−i2π√2 k M(10)ωkδk
X X
( i2π)2 1 ( i2π)3 1
+ −2! √22 k (M(20)ωkω−k+M(22)γkγ−k)+ −3! √23
X
( i2π)4
× (M(30)ωk1ωk2ωk3 +3M(32)ωk1γk2γk3)δk1+k2+k3 + −4!
k1X,k2,k3
1
×√24 k1X,...,k4(M(40)ωk1ωk2ωk3ωk4 +6M(42)ωk1ωk2γk3γk4 +M(44)
3
( i2π)5 1
×γk1γk2γk3γk4)δk1+...+k4 + −5! √25 k1X,...,k5(M(50)ωk1ωk2ωk3
×ωk4ωk5 +10M(52)ωk1ωk2ωk3γk4γk5 +5M(54)ωk1γk2γk3γk4γk5)
( i2π)6 1
×δk1+...+k5 + −6! √26 k1X,...,k6(M(60)ωk1ωk2ωk3ωk4ωk5ωk6
+15M(62)ωk1ωk2ωk3ωk4γk5γk6 +15M(64)ωk1ωk2γk3γk4γk5γk6)
×δk1+...+k6 +...}. (4)
In (4) the cumulants M(in) with i =0 are connected with the nth structure factors of the RS [23]:
n n
M(0) = N S .
n h i n
Structure factors S (0) with n > 2 can be obtained from S (0) by means of a chain of equations for correlation
n 2
functions [29]. Cumulants with i =0 can be expressed in terms of M(0) (see also formulae (4.8) in [23]):
n n
6
M(2) = M(0) , M(4) =3M(0) 2M(0) ,
n n−1 n n−2− n−3
M(6) = 15M(0) 30M(0) +16M(0) . (5)
n n−3− n−4 n−5
First we integrate in (2) over the variables ρk and ωk. As a result, we have
M(0) β
Ξ=Ξ0expn≥1 nn! hn0Z (dc)exp −V k Φ˜C(k)ckc−k!J(c,h0), (6)
X X
 
where
( i2π)2 1 hn
J(c,h0) = Z (dγ)expi2π k γkck+ −2! √22 k γkγ−kn≥0M(n2+)2n0!
X X X

( i2π)4 1 hn
+ −4! √24 k1X,...,k4γk1γk2γk3γk4δk1+...+k4nX≥0M(n4+)4n0!, (7)

βµ
1
hk=0 =h0 = (8)
√2
and condition (3) has the form:
∂lnΞ
1
= N . (9)
∂h h i
0
Expressions(6)-(7) do not include the ”field” variablehk with k=0. As one can see below, this fact will give rise to
6
the Landau type free energy of the RPM in the vicinity of the GL critical point.
Performing in (6)-(7) integration over ck we obtain (up to γ4)
πV ( i2π)2 1
Ξ1 = Yk sβ|Φ˜C(k)|Z (dγ)exp(F(h0)+ −2! Xk γkγ−k2VβΦ˜C(k)
β ∂ (h ) ( i2π)4 1
×(cid:18)1+ V Φ˜C(k) F∂h00 (cid:19)+ −4! √24 k1X,...,k4γk1γk2γk3γk4
∂2 (h ) ∂ (h )
0 0
×δk1+...+k4 3 ∂Fh2 −2 F∂h , (10)
(cid:18) 0 0 (cid:19)(cid:27)
where
M(0) ∂ (h ) M(0)
(h )= n hn, F 0 = n hn−1, (11)
F 0 n! 0 ∂h (n 1)! 0
0
n≥1 n≥1 −
X X
4
∂2 (h ) M(0)
F 0 = n hn−2 (12)
∂h2 (n 2)! 0
0 n≥2 −
X
and formulas (5) are used for M(in).
n
Itshouldbe pointoutthatthis integrationmakessenseforapositivefunctionΦ˜ (k). InourstudyΦ˜ (k)(see (1))
C C
can change the sign and the transition from (6)-(7) to (10) has to be regardedas an algebraic formal way of deriving
the expansion in terms of h0 and γk solely. As a result, in (10) in the exponent we have the expression in terms of
the two fluctuating fields,namely, the fluctuating field hk=0 conjugate to the fluctuating total density ρk=0 and the
fluctuating field γk (for all of k) conjugate to the fluctuating charge density ck.
a. Equation for the chemical potential. Restricting our consideration in (7) to the second power of γk and as-
suming ∂F∂h(h00) >0 (M(n2+)2hnn0! >0), we integrate in (6)-(7) over γk and ck. As a result, we obtain for Ξ1
−1/2
β ∂ (h )
Ξ = 1+ Φ˜ (k) F 0 exp( (h )). (13)
1 C 0
V ∂h F
k (cid:20) 0 (cid:21)
Y
Using (9) and (13) we can write the following equation for the chemical potential
1 βΦ˜ (k) ∂F(h0) M(0) 1M(0)
V C = ∂h0 − 1 =h 1 3 h +... , (14)
2Xk 1+ VβΦ˜C(k)∂F∂h(h00) ∂2∂Fh(20h0) 0 − 2M(20) 0 !
where h , (h ), ∂F(h0) and ∂2F(h0) are given by (8) and (11)-(12), respectively.
0 F 0 ∂h0 ∂h20
b. Phase diagram of the RPM in the MF approximation. We solve equation (14) in the simplest approximation.
Neglecting terms h2, h3 and etc. in the right hand side of (14) and setting
0 0
∂ (h )
F 0 =M(0) (15)
∂h 1
0
in the left hand side of (14) we obtain for µ (=µ )
1,+ 1,−
1
µ = a˜(β), (16)
1,+
−2β
where
βΦ˜ (k)
a˜(β)= V C . (17)
− 1+ βΦ˜ (k)M(0)
Xk V C 1
On the other hand, we can get the same result for µ taking into account the terms proportionalto γ2 and γ2h
1,+ 0
in the exponent of (7) and setting M(2) 0 in the final result (M(2) is the coefficient of γ2h ). Therefore, we may
3 ≡ 3 0
say that µ given by (16)-(17) is obtained in the MF approximation.
1,+
The full chemical potential µ is equal to [22]:
+
µ =µ +µ ,
+ 0,+ 1,+
where µ (=µ ) is the chemical potential of a one-component hard sphere system.
0,+ 0,−
The equationβρ∂µ+ =0,where ρ is the totalnumber density, givesthe spinodalofthe RPMin the approximation
∂ρ
considered. Using(1)forΦ˜ (k)andthePerkus-Yevickapproximationforthe RS,theequationforthespinodalcurve
C
can be written as
∞ x2(sinx)2dx π (1+2η)2
= (18)
(x3T∗+24ηsinx)2 24η (1 η)4
Z0 −
The phase diagram calculated from (18) is shown in Fig. 1. The obtained data for the GL critical point are
T∗ 0.084 (T∗ =1/β∗) and η 0.005.
c ≃ c c c ≃
Itisevidentthatintheconsideredapproximation(correspondingtotheMFtheory)wegettheoverestimatedvalue
forthe GL criticaltemperature andthe underestimatedvalue forthe criticaldensity. But, incontrasttothe previous
results, the spinodal curve changes its run (at η 0.047) and then it directs to the higher temperature. The second
≃
positive slop of the spinodal indicates another type of the phase instability appearing in the RPM (similarly to the
λ-line in a symmetricalbinary fluid). Below,we shallcalculate the phase diagramofthe RPM taking into accountin
(7) the terms of the higher orders.
5
III. CRITICALITY OF THE RPM: BEYOND THE MF THEORY
Now let us rewrite (6)-(7) as
β
Ξ1 = exp( (h0)) (dc)exp Φ˜C(k)ckc−k
F Z −V k !
X
( i2π)2 1
×Z (dγ)exp(i2π k γkck+ −2! √22 k γkγ−kM(10)
X X
( i2π)2 1 ∂ (h ) ( i2π)4 1
+ −2! √22 k γkγ−k(cid:18) F∂h00 −M(10)(cid:19)+ −4! √24
X
∂2 (h ) ∂ (h )
0 0
× γk1γk2γk3γk4 3 ∂Fh2 −2 F∂h δk1+...+k4 (19)
k1X,...,k4 (cid:18) 0 0 (cid:19) 
and the notations in (19) are the same as those in the previous section. 
a. Gaussian approximation. First, we restrict our attention to the second powers of variables h0 and γk which
corresponds to neglecting the terms proportionalto h30, γkγ−kh0, γkγ−kh20, h40, etc. After integration in (19) over γk
we obtain
Ξ = exp( (h )) πM(0) −1/2 (dc)exp ckc−k
1 F 0 Yk (cid:16) 1 (cid:17) Z (−Xk M(10)
β
1+ Φ˜ (k)M(0) . (20)
× V C 1
(cid:18) (cid:19)(cid:27)
As is seen from (20), the equality
β
1+ Φ˜ (k)M(0) =0 (21)
V C 1
holds at some values of the wave-vector k, temperature and density. Equation (21) determines the boundary of
stability connected with the charge fluctuations (the field variable γk is conjugate to the CV ck):
sinx 1
T∗ = 24η , T∗ = , (22)
− x3 β∗
or
cosx∗
T∗(x∗,η)= 8η , (23)
c − x∗2
where x∗ is determined from the condition
x∗
tanx∗ = , (24)
3
which yields x∗ 4.0783. Substituting x∗ in (20) we obtain the boundary of stability with respect to fluctuations of
≃
the local charge density
T∗(x=x∗) 0.285η. (25)
c ≃
Asimilarresult(foranotherchoiceofinteractioninsidethehardcore)wasobtainedin[9,10]withintheframeworkof
the field-theoretical approach. The possibility of the charge-ordering transition in the continuous-space RPM model
was discussed therein.
It is worth noting that the RPM does not demonstrate the GL phase transition in the Gaussian approximation.
In order to obtain the GL spinodal curve we should take into consideration the terms of the order higher than the
second one (γ2h, γ2h2, etc.).
6
b. Beyond the Gaussian approximation(model ϕ4). We restrict our consideration in (19) to the terms of the
fourth order. In this case Ξ has the form:
β
Ξ = Ξ0exp( (h0)) (dc)exp Φ˜C(k)ckc−k (dγ)exp i2π
F Z −V k !Z {
X
( i2π)2
× k γkck+ −2!√22 k γkγ−k M(10)+h0M(20)+h20M(30)
X X (cid:16) (cid:17)
( i2π)4
+ −4!√24 k1X,...,k4γk1γk2γk3γk4(cid:16)3M(20)−2M(10)(cid:17)δk1+...+k4, (26)
where the summation in (11) over n is restricted to 4. 
Now we follow the programme proposed in [16, 23] for a two-component fluid system. First, we separate the two
types of variables: the essential variables (which include the variable connected with the order parameter) and the
non-essential variables. Then, integrating over the non-essential variables with the Gaussian density measure, we
construct the basic density measure (the GLW Hamiltonian) with respect to the essential variables.
For the RPM in the vicinity of the GL critical point the variable h (conjugate to the CV ρ ) turns out to be the
0 0
essential variable [17]. Thus, we can present (26) as
−1/2 β
Ξ1 = Yk (cid:16)πM(10)(cid:17) exp(F(h0))Z (dc)exp −V Xk Φ˜C(k)ckc−k!
1
×(cid:18)1+Aˆ+ 2!Aˆ2+...(cid:19)exp − k ckc−k/M(10)!, (27)
X
where
∂2
ˆ= h M(0)+h2M(0) .
A 0 2 0 3 ∂ckc−k
(cid:16) (cid:17)
After integration in (27) we get
1 hn
Ξ = exp 0 , (28)
1 n
Yk 1+βhNViΦ˜C(k) nX≥1M n!
q  
where
=M(0)+∆M ,
Mn n n
∆Mn are the corrections obtained as the result of integration over variables ck:
1 1
∆M = M(0)a˜(β), ∆M = M(0)a˜(β),
1 2 2 2 2 3
∆M 0, ∆M 0. (29)
3 4
≡ ≡
and a˜(β) is given by (17).
It is worth noting that the non-zero corrections ∆M (see (29)) are those which include only one sum over k.
n
Next, the shift is carried out in order to eliminate the cubic term in (28)
h =h˜ +∆,
0 0
where ∆= M3. Then (28) has the form
−M4
h˜n
Ξ = exp ˜ 0 , (30)
1 n
C  M n! 
n≥1
X
 
7
where
2
1 1 M(0) (M(0))2
= exp M(0)+ 3 M(0) 3
C Yk 1+βhNViΦ˜C(k)  1 2 M(40)! 2 − 4M(40) !
1q (M(0))3
+ a˜(β) M(0)+ 3 ,
2 2 2(M(0))2!!
4
M(0) (M(0))2 1 (M(0))2
˜ = M(0) 3 M(0) 3 + a˜(β) M(0) 3 ,
M1 1 − M(0) 2 − 3M(0) ! 2 2 − M(0) !
4 4 4
(M(0))2 1
˜ = M(0) 3 + a˜(β)M(0), ˜ =M(0).
M2 2 − 2M(0) 2 3 M4 4
4
In(30)h˜ is the field conjugateto the orderparameterforthe GL criticalpoint(see (1.8)-(1.10)in [21]and(15)-(20)
0
in [17]).
Now we can obtain the grand thermodynamic potential of the RPM in the vicinity of the GL critical point
h˜n
Ω Ω +k T ln = k T ˜ 0 . (31)
0 B B n
− C −  M n! 
n≥1
X
 
The righthandside in(31) hasthe formofthe Landaufree energyexpressedin termsofthe fieldh˜ conjugate to the
0
order parameter. From the equation ˜ =0 we obtain the equation for the GL spinodal curve
2
M
M(0) M(0)
a˜(β)= 2 2 + 3 ,
− M(0) M(0)
3 4
or
2 ∞ x2sinxdx S (0) S (0)
2 3
=2 , (32)
π x3T∗+24ηsinx S (0) − S (0)
Z0 3 4
where S (0) is the nth structure factor of the one-component hard-sphere system at k =0.
n
The phase diagram of the RPM is shown in Fig. 2. The curve with the maximum is the GL spinodal calculated
using(32). ThePercus-YevickapproximationisusedforS (0)(theexpressionsforS (0),S (0)andS (0)aregivenin
2 2 3 4
Appendix B). The straight line calculated by (25) (the Gaussian approximation) corresponds to the charge ordering
phase transition. The GL critical point is located at T∗ = 0.0502 and η = 0.022. While the value for T∗ is in good
c c c
agreement with the recent data of computer simulations [12, 13] (T∗ 0.05), the critical density is underestimated
c ≃
(η 0.04).
c
≃
We can also obtain from (31) (using (9)) the expression for µ
+,1
M(0)a˜(β) 1(M(30))3
1 2 − 3(M(0))2
µ = 4 . (33)
+,1 −2βM(0)+ 1M(0)a˜(β) 1(M(30))2
2 2 3 − 2 M(0)
4
If we set M(0) =0 the expression (33) reduces to (16).
3
It should be pointed out that the above described scheme of integration in the vicinity of the GL critical point
cannot be used in the region close to the line defined by (25) (dotted line in Fig. 2): the variables γk appear to be
the essential variables in this region.
IV. CONCLUSIONS
We use the recently developed approach in the study of the critical behaviour of ionic fluids. For the RPM we
obtain the functional of the GPF in terms of the two fluctuating fields, namely, the field h conjugate to the total
0
density fluctuations and the field γk conjugate to the charge density fluctuations. The phase diagram of the system
8
is calculated in the MF approximation (that corresponds to setting M(in) = 0 for n 3 in the expression for the
n
≥
chemical potential) as well as taking into account the higher powers of the field h . In the both cases the phase
0
diagram demonstrates the GL and charge ordering phase instabilities. In the latter case the obtained value for the
GL critical temperature correlates well with the MC simulation data.
Followingthe scenarioproposedin [16, 17] fora binary fluid we integrateoverγk (non-essentialvariables)with the
Gaussiandensitymeasure. Asaresult,weobtainanexplicitexpressionforthegrandthermodynamicpotentialofthe
RPM in the vicinity of the GL critical point as an expansion in terms of the field conjugate to the order parameter.
The expression implies the classical critical behaviour of the RPM.
Appendix A
Let us considera classicaltwo-componentsystemofinteracting particlesconsisting ofN particles ofspecies a and
a
N particles of species b. The system is in volume V at temperature T.
b
Letusassumethataninteractioninthesystemhasapairwiseadditivecharacter. Theinteractionpotentialbetween
a γ particle at~r and a δ particle at~r may be expressed as a sum of two terms:
i j
U (r )=ψ (r )+φ (r ),
γδ ij γδ ij γδ ij
where ψ (r) is a potential of a short-range repulsion that can be chosen as an interaction between two hard spheres
γδ
σ and σ . φ (r) is an attractive part of the potential which dominates at large distances.
γγ δδ γδ
Let us start with a grand partition function of a two-component continuous system
∞ ∞ zNγ β
γ
Ξ= (dΓ)exp U (r ) , (34)
γδ ij
N ! −2 
NXa=0NXb=0γY=a,b γ Z Xγδ Xij
 
where (dΓ) = dΓ , dΓ = d~rγd~rγ...d~rγ is an element of the configurational space of the γth species; z is
γ Nγ Nγ 1 2 Nγ γ
the fugacity of the γth species: z =exp(βµ′ ), µ′ =µ +β−1ln[(2πm β−1)3/2/h3]; β = 1 , k is the Boltzmann
Q γ γ γ γ γ ′ kBT B
constant, T is temperature; m is mass of the γth species, h is the Planck constant. µ is determined from
γ γ
∂lnΞ
= N ,
∂βµ′ h γi
γ
where N is the average number of the γth species.
γ
h i
Further consideration of the problem is done in the extended phase space: in the phase space of the Cartesian
coordinates of the particles and in the CV phase space. An interaction connected with the repulsion (potential
ψ (r)) is considered in the space of the Cartesian coordinates of the particles. We call this two-component hard-
γδ
spheres system a reference system (RS). The thermodynamic and structural properties of the RS are assumed to be
known. The interaction connected with an attraction (potential φ (r) ) is considered in the CV space. The phase
γδ
space overflowis cancelledby introduction ofa Jacobianofthe transitionto CV. The contributionof the short-range
forces to the long-range interaction screening is ensured by averaging this Jacobian over the RS.
Let us introduce the grand partition function of the RS
∞ ∞ exp(βµγN ) β
Ξ = 0 γ (dΓ)exp ψ (r ) , (35)
0 γδ ij
N ! −2 
NXa=0NXb=0γY=a,b γ Z Xγδ Xij
 
where µγ is the chemical potential of the γth species in the RS.
0
Then the grand partition function (34) can be written as [22, 23]:
Ξ=Ξ Ξ ,
0 1
where Ξ is given in (35). The part of the grand partition function which is defined in the CV phase space has the
0
form of the functional integral:
β
Ξ1 = (dρ)exp[β µγ1ρ0,γ − 2V φ˜γδ(k)ρk,γρ−k,δ]J(ρa,ρb). (36)
Z Xγ Xγδ X~k
9
Here,
1) µγ is a part of the chemical potential of the γ-th species
1
β
µγ =µ µγ + φ˜ (k)
1 γ − 0 2V γγ
k
X
and is determined from the equation
∂lnΞ
1
= N ,
∂βµγ h γi
1
µ is the full chemical potential of the γ-th species;
γ
2)ρk,γ =ρck,γ −iρ~sk,γ is the collective variable of the γ-th species, the indices c and s denote the real part and the
coefficient at the imaginary part of ρk,γ; ρ~ck,γ and ρsk,γ describe the value of k-th fluctuation mode of the number of
γ-th species particles. Each of ρc and ρs takes all the real values from to + . (dρ) is a volume element of
k,γ ~k,γ −∞ ∞
the CV phase space:
′
(dρ)= dρ dρc dρs .
0,γ k,γ k,γ
γ k=6 0
Y Y
The prime means that the product over k is performed in the upper semispace;
3) J(ρ ,ρ ) is the transition Jacobian to the CV averagedon the RS [22, 23]:
a b
b ( i2π)n
J(ρa,ρb) = (dν) exp i2π νk,γρk,γ exp −
Z γ=a " k # n≥1 n! ×
Y X X

Mγ1...γn(k1,...,kn)νk1,γ1...νkn,γn , (37)
γ1X...γnk1X...kn #
where variable νk,γ is conjugated to CV ρk,γ. Mγ1...γn(k1,...,kn) is the n-th cumulant connected with
S (k ,...,k ), the n-particle partial structure factor of the RS, by means of the relation
γ1...γn 1 n
Mγ1...γn(k1,...,kn)= n Nγ1...NγnSγ1...γn(k1,...,kn)δk1+···+kn,
where δk1+···+kn is the Kronecker symbol. p
In general, the dependence of M (k ,...,k ) on wave vectors k ,...,k is complicated. Hereafter we shall
replace M (k ,...,k ) by theirγ1v..a.γlunes1in long-nwave length limit M1 (0,n...,0).
γ1...γn 1 n γ1...γn
4) φ˜ (k) is the Fourier transform of attractive potential φ (r). The behaviour of φ (r) in the region of the core
γδ γδ γδ
r <σ must be determined from the conditions of optimal separation of the interaction.
γδ
We pass in (36) to CV ρk and ck (according to ωk and γk) by means of the orthogonallinear transformation:
√2 √2
ρk = (ρk,a+ρk,b), ck = (ρk,a ρk,b),
2 2 −
√2 √2
ωk = (νk,a+νk,b), γk = (νk,a νk,b).
2 2 −
As a result, we obtain for Ξ [16, 17]
β
Ξ = Ξ0 (dρ)(dc)exp βµ+1ρ0+βµ−1c0− 2V [V˜(k)ρkρ−k+
Z h Xk
W˜(k)ckc−k+U˜(k)ρkc−k] J(ρ,c). (38)
i
Here the following notations are introduced:
10
ρk and ck are the CV connected with total density fluctuation modes and relative density (or concentration)
fluctuation modes in the binary system. (dρ) and (dc) are volume elements of the CV phase space:
′ ′
(dρ)=dρ dρcdρs, (dc)=dc dccdcs.
0 k k 0 k k
k=6 0 k=6 0
Y Y
Functions µ+ and µ− have the form:
1 1
√2 √2
µ+ = (µ +µ ), µ− = (µ µ ) (39)
1 2 1,a 1,b 1 2 1,a− 1,b
V˜(k) = (φ˜ (k)+φ˜ (k)+2φ˜ (k))/2
aa bb ab
W˜(k) = (φ˜ (k)+φ˜ (k) 2φ˜ (k))/2
aa bb ab
−
U˜(k) = (φ˜ (k) φ˜ (k))/2, (40)
aa bb
−
J(ρ,c)= (dω)(dγ)exp i2π (ω ρ +γ c ) J(ω,γ), (41)
k k k k
Z h Xk i
( i2π)n
J(ω,γ) = exp − M(in)(0,...,0)
n! n ×
hnX≥1iXn≥0 k1X...kn
γk ...γk ωk ...ωk . (42)
1 in in+1 n
i
Indexinisusedtoindicatethenumberofvariablesγk inthecumulantexpansion(42). CumulantsMn(in) areexpressed
as linear combinations of the partial cumulants M (see (37)) and are presented for γ ,...,γ = a,b and n 4
γ1...γn 1 n ≤
in [23] (see Appendix B in [23]).
Appendix B
(1 η)4
S (0) = − ,
2 (1+2η)2
(1 η)7(1 7η 6η2)
S (0) = − − − ,
3 (1+2η)5
(1 η)10(1 30η+81η2+140η3+60η4)
S (0) = − − .
4 (1+2η)8
Acknowledgement
The author thanks M.F. Holovko for helpful discussions.
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