Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://chaos.phys.msu.ru/loskutov/PDF/Fulltext_TMPh_fluid.pdf Äàòà èçìåíåíèÿ: Thu Feb 5 11:57:54 2009 Äàòà èíäåêñèðîâàíèÿ: Mon Oct 1 20:06:38 2012 Êîäèðîâêà:

Theoretical and Mathematical Physics, 130(1): 123­135 (2002)

FIELD THEORY ANALYSIS OF CRITICAL BEHAVIOR OF A SYMMETRIC BINARY FLUID
N. V. Brilliantov,1 A. Yu. Loskutov,1 and V. V. Malinin
1

A method is elab orated for constructing an effective field theory Hamiltonian of the Landau­Ginzburg­ Wilson typ e for off-lattice models of binary fluids. We show that all coefficients of the effective Hamiltonian for a symmetric binary fluid can b e expressed in terms of some known characteristics of the model hardsphere fluid, namely, compressibility and its derivatives with resp ect to density. Application of the effective Hamiltonian is demonstrated by an example of determining the curve of critical layering p oints in the mean-field approximation. This curve agrees well with numerical exp eriment results for symmetric binary fluids.

1. Introduction
The mo dern theory of critical phenomena, which is based on the universality hypothesis and (technically) on the renormalization group (RG) metho d, is successfully applied in many divisions of physics ranging from magnetic systems and solutions to pro cesses of self-avoiding walks and percolation [1], [2]. Application of the RG method to critical phenomena was initially developed for magnetic systems (see, e.g., [3] and [4]). The ma jority of investigations were devoted to these systems. The RG metho d was also used to investigate critical properties of fluids, and a number of effective algorithms were elaborated for calculating thermo dynamic quantities in the domain where critical fluctuations play a dominant role [5]­ [9]. These approaches are based on the direct application of the Kadanoff­Wilson recursive scheme (see, e.g., [4]). But this do es not permit the general analysis of critical behavior as visually and simply as do the metho ds based on using the system Hamiltonian in the Landau­Ginzburg­Wilson (LGW) form. This Hamiltonian is written as H= dr a2 2 + B4 4 + B6 6 + ··· + b2 ( )2 ,

where is the magnetization or the order parameter or, as it is also called, the "critical field," and the coefficients a2 , B4 , B6 ,... and b2 characterize the respective dependence of the Hamiltonian (or free energy [4]) on the degree of the field and its spatial gradient. A Hamiltonian of this type is also called a "field theory" Hamiltonian. Knowing such an effective Hamiltonian of a system, one can draw conclusions about the character of its critical behavior and universality class, determine the domain of influence and the magnitude of the contribution of critical fluctuations to thermo dynamic functions, derive simple mean-field estimates for the position of the critical point, and so on. It is therefore very important to find the field theory Hamiltonian for a system possessing critical behavior. Effective field Hamiltonians for fluids were constructed by different metho ds in a number of works. For example, in [10], the Hamiltonian of a fluid was reduced to a field Hamiltonian by coarsening the
1 Moscow State University, Moscow, Russia, e-mail: kolja@gran.phys.msu.su, loskutov@moldyn.phys.msu.su, vitmalin@mail.ru.

Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 130, No. 1, pp. 145­158, January, 2002. Original article submitted March 22, 2001. 0040-5779/02/1301-0123$27.00 c 2002 Plenum Publishing Corp oration
123


spatial scale. In [11], the co efficients of the effective Hamiltonian were found by comparing the values of compressibility, correlation length, and critical amplitudes of the order parameter calculated in the framework of the generalized mean-sphere approximation with the same quantities calculated using the LGW Hamiltonian. In [12], a functional generalization of the Mayer expansion was used. In this case, attractive interactions were considered at the level of the second virial co efficient, and a number of different approximations were used for repulsive interactions. The indicated approximations were used in [12] to calculate the co efficients of the effective LGW Hamiltonian for a one-component fluid. A similar approach, but in combination with somewhat different approximations, was used to calculate the coefficients of the effective Hamiltonian for the so-called restricted primitive electrolyte mo del [12]­[14]. In contrast to all these approaches, which use some approximations, Hubbard and Schofield [15] suggested an exact method for constructing an effective field Hamiltonian of a "real," off-lattice fluid using an integral transformation of the Hubbard­Stratonovich type . Although no co efficients of the effective Hamiltonian were calculated in [15], they convincingly argued that critical behavior of fluids relates to the same universality class as Ising magnets. A further development of the Hubbard­Schofield metho d was suggested in [16] and [17], and it was shown that all co efficients of the effective LGW Hamiltonian for a one-component fluid can be expressed via some known characteristics of the mo del hard-sphere fluid, namely, compressibility and its derivatives with respect to density. In [18], this metho d was used to find the LGW Hamiltonian for Coulomb fluids. In particular, using the RG metho d, it was shown that the unusual critical behavior of such systems is related to the negativity of the co efficient B4 . In this case, it was established that the lattice and off-lattice mo dels have different types of critical behavior in these systems. We note that the Hubbard­Schofield metho d is much simpler than the collective variables metho d, which is also an exact metho d for finding the effective Hamiltonian of a fluid [19]. Unlike one-component systems, two-component fluids can show both the ordinary critical behavior corresponding to the liquid­gas transition and the critical behavior related to the layering into two phases, one enriched by the A component and the other by the B component. In the first case, the order parameter is the difference between the densities of the liquid and gas phases (with uniform compositions), and in the second case, it is the difference between the concentrations of the A and B components in the phases. In the first case, the liquid­gas co existence curve in the temperature­density plane ends at a single critical point, and in the second case, an entire line of critical layering points, usually called a -line,2 appears in this plane. The -line ends at its point of intersection with the liquid­gas co existence curve (see Fig. 1), and this point is called the final critical point (FCP). The critical behavior of systems whose phase diagram has a -line and an FCP has been actively investigated in recent years. The main questions here relate to the universality class and the character of critical behavior along the -line and at the FCP [20]­[23]. For example, the nonclassical character of critical behavior near the FCP was revealed in the framework of the spherical mo del [20]. Similar behavior on the -line was established theoretically [21], [22] and numerically using the Monte Carlo metho d [23]. In [24], the phase behavior of a symmetric binary fluid (i.e., a fluid consisting of identical A and B components differing only in the interaction potentials for particles of different types) was investigated both numerically and with the phenomenological expansion of the free energy with respect to the order parameter. In this case, the dependence of phase diagrams on the parameter equal to the ratio of the interaction energy for particles of one type to that for particles of different types was studied. A qualitative field theory analysis of the critical behavior of a system with a Hamiltonian depending on two fluctuating fields (one of which is regular at the critical point) was performed in [25], where the Hubbard­
2 For mixtures, it is also necessary to introduce the axis of relative concentrations in addition to density and temp erature. Then a coexistence curve in the density­composition coordinates with the related critical point corresponds to each temperature. This precisely leads to the appearance of the line of critical layering points (the -line).

124


Fig. 1.

Typical structure of phase diagrams for two-comp onent systems having a -line (3) and

an FCP (2). Point 1 is the critical gas­liquid transition p oint. The resp ective domains I, I I, and I I I corresp ond to the gas phase, the homogeneous mixture of particles of typ es A and B , and the coexisting phases enriched by one (A) or the other (B ) comp onent.

Stratonovich transformation for the spin Hamiltonian corresponding to the lattice gas of a symmetric binary mixture was used. It was shown that the critical behavior near the FCP is equivalent to that along the -line. In [26], the collective variables metho d was used to study the critical liquid­gas transition point for a binary fluid. In this paper, the critical behavior of a symmetric binary fluid is investigated by constructing an effective field theory Hamiltonian of the LGW type. We use the Hubbard­Schofield transformation of the original Hamiltonian of the fluid [16], [17]. In contrast to the similar Hubbard­Stratonovich transformation, this transformation do es not require the existence of the inverse operator for the part of the interaction potential corresponding to the interactions of the impenetrable particle cores. In particular, this permits using off-lattice mo dels. The singular part of this potential is considered separately and is a basis for calculating the co efficients of the effective Hamiltonian. In the framework of this approach, an effective field theory Hamiltonian can be constructed with explicit expressions for the co efficients for field powers of any order. As an example of applying the effective Hamiltonian, we determine the curve of critical layering points (the -line), which agrees well with numerical experiment results [24].

2. Derivation of the effective LGW Hamiltonian
The particle interaction potential U (r)ab in a binary mixture can be represented as a sum of two parts, ab U (r)ab = Ur (r)ab + J (r)ab , where Ur and J ab describe the respective purely repulsive and purely attractive interactions, a = A, B , and b = A, B . The suggested approach is valid for any interaction potential representable as a sum of repulsive and attractive parts. To simplify the presentation of the material and also to make it possible to compare predictions of our theory with numerical experiment results [24], we consider an interaction potential of the type of a rectangular potential well. Moreover, we assume that the repulsive interactions in the binary
125


mixture under study are the same for all types of particles, for r < d, 0 for r d,

U

AA r

=U

BB r

=U

AB r

= Uhs (r) =

where d is the diameter of the hard cores of the particles, and the attractive part of the potential has the form J
AA

=J

BB

= J (r) =

-J 0

for r d, for r > d,

J

AB

=

-

for J d, 0 for r > d.

Here, d is the width of the rectangular potential well, J describes the attraction between the particles (the depth of the potential well), and we intro duce the parameter = J AB /J 1 determining the relative interaction force for the particles of one type and different types. We consider the Hamiltonian of the above system (omitting its ideal part), - 1 2
ab Jk na n k a,b k b -k

H=H

hs

1 1 + NA J (0) + NB J (0). 2 2

(1)

Here, Hhs characterizes the repulsive interaction between the hard cores of the particles, and all the other terms in the right-hand side of (1) relate to the attractive part Ha of the Hamiltonian represented via the collective variables, n
a,b k

1 = V

N

a,b

e
j =1

-ikra,b j

,

(2)

where rj are the co ordinates of the j th particle. (The parameters Na and Nb are the numbers of particles of each of the types, a = A, B , and b = A, B .) Here, V is the volume of the system and Jk = J (r)e-ikr dr. The summation in (1) ranges the wave vectors k = {kx ,ky ,kz } with ki = 2li /L (i = x, y , z ), where li are integers and L3 = V . We intro duce the new variables qk = na - nb , k k sk = n a + n b , k k (3)

for which the attractive part of the Hamiltonian has the form3 Ha = - 1 2 Jk Jk (1 - )qk q-k + (1 + )sk s 2 2 1 1 + NA J (0) + NB J (0). 2 2

-k

(4)

k

Then it becomes possible to write the configuration integral of the system in question [15], [16], 2 Jk (1 - )qk q-k +(1+ )sk s 2
-J (0)(NA +NB )/2

Q=

exp

-k r

Qr e

,

(5)

k

where = 1/kB T , kB is the Boltzmann constant, and T is the temperature. The expression Qr is the configuration integral of the base system of hard spheres, and ( · ) r = Q-1 drN ( · ) denotes the average r
3 In the general case of a binary system with an attractive p otential of an arbitrary form (for which it is not allowable to AA AB write J AB = J AA , 1), the expressions Jk (1 ± ) in (4) should be replaced with Jk ± Jk .

126


over this system. We note that the configuration integral Qr is a regular function in the phase-space domain under consideration because the base hard-sphere system has no critical layering points; therefore, only the first term in (5) determines the critical behavior of the system. In accordance with the Hubbard­Schofield scheme [15], i.e., using the identity
+ 2

e

a2 x2 /2

1 = 2a

e
-

-y 2 /(2a2 )+xy

dy ,

we bring the configuration integral Q to the form [16] Q
k

dk
p

dp exp -

1

k

k -k 1 - (1 - )Jk
-p

p

p -p (1 + )Jp

â

â exp
k

k q-k +
p

p s

,
r

(6)

where the factor that is regular at the critical point and do es not affect the further analysis has been omitted. The integration in (6) is performed under the condition that -k = k and -p = p . (Here, k and p are the respective complex conjugates of k and p .) We assume that the critical point for the gas­liquid transition lies sufficiently far from the line of critical layering points (see Fig. 1). It is clear that in this case, only the field k conjugate to the variable qk = na - nb is critical, whereas the field p conjugate to sp = na + nb is regular.4 Therefore, applying the p p k k cumulant expansion for exp k q-k + p p s-p r [28], we retain the terms up to the fourth degree k with respect to the field variable in resulting expression (6) (because the co efficient in 2 vanishes in the neighborho o d of the -line) and up to the second degree with respect to (because the co efficient of 2 is always nonzero in the domain under consideration). This results in the relation Q
k 4

dk
p

dp exp - 1

1

k

k -k + (1 - )Jk

p

F (p) p + V G(p1 , p2 ) p1 p V
2

+
n=1

V

n/2-1 k1 ,...,k
n

un (k1 ,..., kn )k1 ··· kn +
p1 ,p

2

(7)

with the expressions for the co efficients F and G F (p) = v1 (p)+2 V +4
k1 ,...,k
3

k

w11 (k, p) k +3 V

k1 ,k

w21 (k1 , k2 , p) k1 k2 + V
2

w31 (k1 ,..., k3 , p) k1 ··· k3 + V 3/2 w41 (k1 ,..., k4 , p) k1 ··· k4 , V2
4

+5
k1 ,...,k

4 It was shown in [27] that the order parameter and its conjugate field in the corresp onding integral transformation have the same critical behavior.

127


1 G(p1 , p2 ) =- + v2 (p1 , p2 )+ 3 V (1 + )Jp +6
k1 ,k
2

k

w12 (k, p1 , p2 ) k + V

w22 (k1 , k2 , p1 , p2 ) k1 k2 + V w32 (k1 ,... , k3 , p1 , p2 ) k1 ··· k3 + V 3/2
3

+10
k1 ,...,k

+15
k1 ,...,k
4

w42 (k1 ,... , k4 , p1 , p2 ) k1 ··· k4 , V2

where un (k1 ,..., kn ) = V
n/2-1

n! V m!

qk1 ,...,qk

n

c ,r

, , qk1 ,...,qkn ,sp1 ,... ,sp .

m/2-1

vm (p1 ,... , pm ) =

sk1 ,... ,sp V

m

c ,r

(n+m)/2-1

wnm (k1 ,..., kn , p1 ,..., pm ) =

(n + m)!

m

c ,r

Here · c,r denotes the cumulant mean calculated for the base system. Using the definitions of equilibrium correlation functions of a fluid [29] and of cumulant means [28], we can express the co efficients of the powers of and in (7) explicitly in terms of the Fourier transforms of the correlation functions h1 ,h2 ,...,hn of the base hard-sphere system, which are defined as h1 (r) (r), h2 (r1 , r2 ) g2 (r1 , r2 ) - 1, (8) ...,

h3 (r1 , r2 , r3 ) g3 (r1 , r2 , r3 ) - g2 (r1 , r2 ) - g2 (r1 , r3 ) - g2 (r2 , r3 )+2,

where gl (r1 ,... , rl ) are the l-particle equilibrium correlation functions of the base system. As the simplest example, we consider the main stages in calculating the function w11 (k, p). Using the definitions of cumulant means [28], w11 (k, p) = we obtain qk s
p c ,r

1 qk s 2!
a pr

p c ,r

=

1 qk s 2!

pr

- qk

r

s

pr

,

(9)

= na n k

-n

a kr

n

a pr

+ na n k

b pr

-n

a kr

n

b pr

- (10)

- nb n k

a pr

+n

b kr

n

a pr

- nb n k

b pr

+n

b kr

n

b pr

for the variables qk and sp using expressions (2) and (3). The definition of a binary symmetric fluid implies that the averaging in calculating means of the form n1 ,n2 ,... ,n n , where , ,..., = A, B , is k k k performed with respect to the Hamiltonian of the base system of hard-sphere particles of diameter d. (This system is the same for particles of types A and B .) We now take representation (2) and the definition of equilibrium correlation functions [29] into account. For example, this gives na n k
a pr

-n

a kr

n

a pr

=

a V - 2 a V

e

-ikr1 -ipr

1

dr1 + e

2 a V

e

-ikr1 -ipr2

g2 (r1 , r2 ) dr1 dr2 -

e

-ikr1

dr1

-ipr2

dr2 =

=
128

k +p , 0

~ {2 h2 (k)+ a } a


for the Fourier transform, where a = NA /V and b = NB /V are the respective particle densities for types ~ A and B , hl denotes the Fourier transform of hl , and the relation k,0 V = eikr dr expressing the Kronecker delta for a discrete argument is used. Calculating analogously for the other terms in (10), we obtain the final expression for w11 (k, p) (see (9)),
k +p , 0

w11 (k, p) =

2!

~ h2 (k)(a + b )(a - b )+(a - b ) .

The co efficients in (7) can be calculated similarly up to an arbitrary order. For example, u1 = u2 = u3 =
k1 ,0

{ a - b } , ~ 2 (k1 )(a - b )2 +(a + b ) , h ~ 3 (k1 , k2 )(a - b )3 +(a - b )+ h



k 1 +k 2 , 0

2! 3!

k 1 +k 2 +k 3 , 0

~ ~ ~ + h2 (k1 )+ h2 (k2 )+ h2 (k3 ) (2 - 2 ) , a b w21 =
k 1 +k 2 +p 1 , 0

3!

~ 3 (k1 , k2 )(a - b )2 (a + b )+(a + b )+ h

~ ~ ~ + h2 (k1 )+ h2 (k2 ) (a - b )2 + h2 (p1 )(a + b )2 , and so on. It follows from the variable are identically zero equal to the average number of Integrating with respect to integral of the system: Q
k

structure of the co efficients that all co efficients of o dd powers of the field for a symmetric system (where the average number of type-A particles is type-B particles, i.e., a = b ). the set of variables {p } in (7), we obtain the relation for the configuration

dk exp -

1

k

k -k + (1 - )Jk

4

1 V
n/2-1 k1 ,...,k
n

un (k1 ,..., kn )k1 ··· k

n

â

n=1

â exp -
p1 ,p
2

1 F (p1 )G-1 (p1 , p2 )F (p2 ) - log[det G(p1 , p2 )] , 4

(11)

where G-1 (p1 , p2 ) is the inverse matrix of G(p1 , p2 ). Assuming that the critical liquid­gas transition point is lo cated far from the -line, we can neglect the fluctuations of the full density = a + b of the fluid and keep only the terms corresponding to p1 = 0 and p2 = 0 in (11). We next expand the functions F (0) and G(0, 0) in (11) up to the fourth-order terms with respect to the field variable k , which corresponds to the usual LGW representation in analyzing the critical behavior of a system in the three-dimensional space. Then collecting the terms in like powers of k , we find Q
k

k e

-H

with the effective Hamiltonian H given by the formula B4 k1 ··· k4 + V
4

H=
k1 ,k
2

B2 k1 k2 +
k1 ,...,k

k1 ,k

D4 k1 k V
2

2

k3 k4 .
k3 ,k
4

(12)

129


(The expressions for the co efficients in the effective Hamiltonian are given below.) We are interested in the behavior of the system in the vicinity of the curve of critical layering points (the -line), i.e., in the domain where the ma jor role is played by large-scale fluctuations of the density difference, which corresponds to fluctuations with small wave vectors. We therefore consider the expansion of the Hamiltonian co efficients for small k . We note that the LGW Hamiltonian written in terms of the Fourier transforms of field variables contains no terms with powers of k higher than k 2 . (This is equivalent to the absence of gradient terms of an order higher than two in the LGW Hamiltonian [4].) Moreover, the only term of the order of k 2 is proportional to k 2 k -k . Hence, for n > 2, only the zeroth-order terms should be kept in expansion (12) for the co efficients of the effective Hamiltonian, and we represent the co efficient B2 in the form B2 = a2 + b2 k 2 . ~ ~ We use the expansion for the correlation functions hl (k ) = ~ l (0) - k 2 hl (0) with small k . (In this h ~ ~ ~ case, hl (0) = hl (0,... , 0).) The value of h2 (0) can be expressed via the isothermal compressibility r = -1 ~ (/ Pr ) of the base hard-sphere system in the form 1 + h2 (0) = kB Tr z0 , where z0 is the reduced density and Pr and = a + b are the corresponding pressure and density in the base system. Using the well-known formulas relating the correlation functions of different orders [29] l gl = l lgl +



2

drl

+1

(g

l+1

- gl ) ,

~ we derive a formula for the zeroth moments of the functions hl (0) [16],
2



l~ ~ hl (0) = l l~ l (0) + h h

l+1

(0) .

(13)

~ Recursive application of Eq. (13) to itself can give an expression for each of the functions hl (0) in terms of the reduced compressibility z0 and its derivatives with respect to density, for example, ~ h2 (0) = ~ h3 (0) = ~ h4 (0) =
-1 -2 -3

(z0 - 1),
2 (z0 - 3z0 +2+ z0 z1 ), 3 2 2 2 2 (z0 - 6z0 +11z0 - 6+4z0 z1 - 6z0 z1 + z0 z1 + z0 z2 ),

where z1 = z0 / and z2 = 2 2 z0 / 2 . We thus reach the conclusion that all co efficients of effective Hamiltonian (12) can also be expressed in terms of the isothermal compressibility r of the base system and its derivatives with respect to density. Using the very exact Carnahan­Starling state equation [30] for the hard-sphere system, Fex N
,hs

=

4 - 3 2 , (1 - )2

where = (/6)d3 is the packing co efficient, and differentiating twice with respect to density, we obtain z0 = (1 - )4 (1 + 4 +4 2 - 4 3 + 4 )-1 . Next, we express the effective Hamiltonian via the field depending on the spatial co ordinate. For this, it is necessary to transform the variables k to the field variables (r). Under this transformation, integration with respect to the set of variables {k } is replaced with integration with respect to the field
130


(r), and the term new variables a2 = -1/3 FCP. We use c final expression for -line,

proportional to k 2 k -k becomes the one proportional to [ (r)]2 . We intro duce 1/3 c a2 , b2 = c b2 , B4 = c B4 , and D4 = c D4 , where c is the critical density at as the length-scale factor. Performing all the necessary transformations, we obtain the effective LGW Hamiltonian of a symmetric binary fluid in the neighborho o d of

the the the the

H= where
2

dr a2 2 + B4 4 + D4 2

2

+ b2 ( )2 ,

=V

-1

2 (x) dx and the co efficients a2 , b2 , B4 ,and D4 have the forms

a2 c =

3w21 (0) 1 32 w22 (0) + - - u2 (0), 2G0 (1 - )J0 2G2 0 3 2 3 J0 2 w22 (0) - 2G w21 (0) + (1 - )J 2 , 2G0 0 0 (14)

b2

1/3 c

=

5w41 (0) 152 w42 (0) B4 c = - - u4 (0), 2G0 4G2 0 D 4 c = In (14), we have G0 = - v2 (0) = u2 (0) = 1 + v2 (0), (1 + )J0 (h2 (0) + 1), 2! , 2! 9 2 w22 (0) . w21 (0) - 4G0 G0
2

(15)

~ u4 (0) = (3h2 (0) + 1), 4! w41 (0) = w42 (0) = ~ ~ (32 h3 (0) + 7h2 (0) + 1), 5! ~ ~ h 33 h4 (0) + 162 ~ 3 (0) + 15h2 (0) + 1 , 6!

and b2 has been calculated using the representation v2 = v2 (0) - k 2 v2 (0), wnm = wnm (0) - k 2 wnm (0), and Jk = J0 - k 2 J0 . The zeroth and second moments of the attractive part of the interaction potential are found from the relations J0 = 1 J0 = 6 with the effective temperature T = kB T/J .
131

J (r) dr =

4d3 3 , 3T (16)

2 d2 J0 r2 J (r) dr = 10


The zeroth and second moments of w21 and w22 can be expressed in terms of those of the correlation ~ functions hl (0), w21 (0) = w21 (0) = ~ h2 (0) + 1 , 3!

2 ~ h (0), 3! 2 2~ ~ h3 (0) + 3h2 (0) + 1 , w22 (0) = 4! 2~ ~ w22 (0) = h3 (0) + 3h2 (0) . 4! ~ The function h2 (k) is representable via the Fourier transform of the direct correlation function c2 (k) ~ of the base system, h2 (k) = c2 (k)/(1 - c2 (k)). Expanding the Fourier transform c2 (k) as a Taylor series, ~ ~ ~ c2 (k) = c2 (0) - c2 (0)k 2 + ... , we obtain ~ ~ ~ ~ ~ h2 (k) = ~ 2 (0) - k 2 c2 (0)(1 + h2 (0))2 , h ~ ~ ~ whence follows the relation h2 (0) = c2 (0)(1 + h2 (0))2 . To calculate c2 (0), we use the Wertheim­Thiele an~ alytic solution of the Percus­Yevick equation for the direct correlation function of the hard-sphere system.5 This results in the expression d5 (16 - 11 +4 2 )(1 - )- 120

c2 (0) = ~ for c2 (0). ~

1 6

r2 c2 (r) dr = -

4

~ ~ In contrast to h2 (0), the expression h3 (0) cannot be written in terms of the compressibility of the base system. For ~ 3 (0), we have the formula h (kr2 )2 (kr1 )2 + - (kr1 )(kr2 ) dr1 dr2 , 2 2

~ h3 (0)k 2 =

h3 (r1 ,r2 ,r12 )

(17)

~ where r12 = |r1 - r2 |. Formula (17) shows that to calculate h3 (0), it is than the zeroth) moment of the three-particle correlation function. By be written in terms of the three-particle function g3 (r1 ,r2 ,r12 ), whose But the calculations can be performed using some approximations, e.g

required to know the second (rather (8), the expression h3 (r1 ,r2 ,r12 ) can analytic representation is unknown. ., the superpositional approximation

g3 (r1 ,r2 ,r12 ) = g2 (r1 )g2 (r2 )g2 (r12 ).

(18)

5 According to [31] and [32], the direct correlation function of the hard-sphere system has the form c (r ) = 0 for r 1 2 and c2 (r ) = (1 - )-4 -(1 + 2)2 +6(1 + /2)2 r - (1 + 2)2 r 3 /2 for r < 1, where r = r/d and d is the diameter of the hard spheres.

132


Fig. 2. are

Comparison of the data obtained for the -line theoretically (solid line) with the results of

exp erimental numerical calculation using the Monte Carlo method (dashed lines). The parameter values = 0.72 and = 1.5.

Numerical calculations of the coefficient b2 using approximation (18) showed that b2 is positive in the domain we are interested in for any values of the system parameters. We have thus constructed the effective LGW Hamiltonian for a symmetric binary fluid. Using a similar argument, we can obtain analytic expressions for the co efficients of an arbitrary order with respect to the field variable. We note that the resulting Hamiltonian involves two different fourth-order terms with respect to the field variable that are proportional to the corresponding expressions 4 and 2 2 . As an example of applying the effective field theory Hamiltonian, we consider the mean-field condition for the critical point according to which the co efficient in the second power with respect to the field must vanish as the temperature tends to the critical value, i.e., a2 = 0. We rewrite expression (14) for a2 as 1 3 w22 (0) = - . w21 (0) - 2G0 G0 (1 - )J0

(19)

Formula (19) corresponds to the actual mean-field condition under which the field is assumed to be constant 2 throughout the system volume, i.e., (x) = 0 and 2 = 0 . In this case, the Hamiltonian (or the free2 4 energy density [4]) has the form H = a2 0 +(B4 + D4 )0 . Use can also be made of a more complicated selfconsistent condition imposed on the critical point taking the field fluctuation in the Gaussian approximation, a2 + D4 2 c = 0, into account. (Here 2 c is calculated at the critical point.) But this approximation is not as simple and convenient as (19). Using (19), (15), and (16), we calculated the dependence of the critical density on the critical temperature along the critical layering curve, i.e., the -line, in the case of particle interaction described by a rectangular potential well. The calculation results are presented in Fig. 2, where the -line found numerically by the Monte Carlo metho d for the parameter values = 0.72 and = 1.5 [24] is also shown. As is seen, resulting mean-field relation (19) agrees well with the numerical experiment. It turned out that the co efficient B4 can take both positive and negative values along the -line. In this case, the co efficient B6 is always positive. (We do not present the analytic expression for this co efficient because it is very cumbersome.)
133


3. Conclusion
In this paper, a method is suggested for constructing an effective field theory Hamiltonian of the LGW type for off-lattice mo dels of binary fluids. It is based on using an integral transformation of variables of the Hubbard­Stratonovich type. In contrast to the Hubbard­Stratonovich transformation, however, this metho d do es not require the existence of the inverse operator for the singular part of the potential describing the interaction between the impenetrable particle cores. This permits considering off-lattice fluid models with a specific form of the interaction potential for hard cores. We have shown that for a symmetric binary fluid, all coefficients of the effective Hamiltonian can be expressed in terms of the compressibility and its derivatives with respect to the density of the "support" system possessing only repulsive interactions. In the case of repulsive interactions mo deled using hardsphere interactions, there exist very accurate analytic expressions for these derivatives. Calculations were performed for a symmetric binary mixture with an interaction potential of the type of a rectangular potential well, and several first co efficients were determined for the effective LGW Hamiltonian. The curve of critical layering points (the -line) was found in the mean-field approximation, and it turned out to agree well with the numerical experiment results obtained using the Monte Carlo metho d. Acknowledgments. This work was supported in part by INTAS-OPEN (Grant No. 97-1094). REFERENCES
1. H. E. Stanley, Introduction to Phase Transitions and Critical Phenomena, Oxford Univ. Press, New York (1971). 2. N. Goldenfeld, Lectures on Phase Transitions and the Renormalization Group , Addison-Wesley, New York (1992). 3. Y. M. Ivanchenko and A. A. Lisyansky, Physics of Critical Fluctuations , Springer, New York (1995). 4. Shang-keng Ma, Modern Theory of Critical Phenomena, Benjamin, Reading, Mass. (1976). 5. J. A. White and S. Zhang, J. Chem. Phys., 99, 2012 (1993). 6. J. A. White and S. Zhang, J. Chem. Phys., 103, 1922 (1995). 7. L. Reatto and A. Parola, J. Phys.: Cond. Mat., 8, 9221 (1996). 8. M. Tau, A. Parola, D. Pini, and L. Reatto, Phys. Rev. E , 52, 2644 (1995). 9. A. Parola and L. Reatto, Adv. Phys., 44, 211 (1995). 10. C. Vause and J. Sak, Phys. Rev. A, 21, 2099 (1980). 11. R. J. Leote de Carvalho and R. Evans, J. Phys.: Cond. Mat., 7, L575 (1995). 12. M. E. Fisher and B. P. Lee, Phys. Rev. Lett., 77, 3561 (1996). 13. Y. Levin and M. E. Fisher, Physica A, 225, 164 (1996). 14. M. E. Fisher, J. Phys.: Cond. Mat., 8, 9103 (1996). 15. J. Hubbard and P. Schofield, Phys. Lett. A, 40, 245 (1972). 16. N. V. Brilliantov, Phys. Rev. E , 58, 2628 (1998). 17. N. V. Brilliantov and J. Valleau, J. Chem. Phys., 108, 1123 (1998). 18. N. V. Brilliantov, C. Bagnuls, and C. Bervillier, Phys. Lett. A, 245, 274 (1998). 19. I. R. Yukhnovskii, Phase Transitions of the Second Order : The Collective Variables Method [in Russian], Naukova Dumka, Kiev (1985); English transl., World Scientific, Singap ore (1987). 20. M. C. Barb osa, Physica A, 177, 153 (1991). 21. M. E. Fisher and M. C. Barb osa, Phys. Rev. B , 43, 11177 (1991). 22. Y. C. Kim, M. E. Fisher, and M. C. Barb osa, "The isothermal binodal curves near a critical endp oint," condmat/0102195 (2001). 23. N. B. Wilding, Phys. Rev. Lett., 78, 1488 (1997). 24. N. B. Wilding, F. Schmid, and P. Nielaba, Phys. Rev. E , 58, 2201 (1998). 25. H. W. Diehl, Physica A, 281, 268 (2000).
134


26. I. R. Yukhnovskii, J. Stat. Phys., 81, 647 (1995); O. V. Patsahan, M. P. Kozlovskii, and R. S. Melnyk, "Ab initio study of the vap our­liquid critical p oint of a symmetrical binary fluid mixture," cond-mat/9907195 (1999). 27. J. J. Binney, N. J. Dowrich, A. J. Fisher, and M. E. J. Newman, The Theory of Critical Phenomena, Clarendon, Oxford (1993). 28. R. Kub o, J. Phys. Soc. Jap., 17, 1100 (1962). 29. C. G. Gray and K. E. Gubbins, Theory of Molecular Fluids , Clarendon, Oxford (1984). 30. N. F. Carnahan and K. E. Starling, J. Chem. Phys., 51, 635 (1969). 31. M. S. Wertheim, Phys. Rev. Lett., 10, No. 8, 321 (1963). 32. E. Thiele, J. Chem. Phys., 39, 474 (1963).

135