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Free energy of dipolar hard spheres: The virial expansion under the presence of an external magnetic field
Physica A 415 (2014) 210–219
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Physica A journal homepage: www.elsevier.com/locate/physa
Free energy of dipolar hard spheres: The virial expansion under the presence of an external magnetic field Ekaterina A. Elfimova ∗ , Tatyana E. Karavaeva, Alexey O. Ivanov Institute of Mathematics and Computer Sciences, Ural Federal University, 51 Lenin Avenue, Ekaterinburg 620000, Russia
highlights • • • • •
The model of dipolar hard spheres in an external magnetic field was considered. The expressions for 2nd (B2) and 3rd (B3) virial coefficients were derived. The formula for B3 for a system in the field is different from the zero-field case. The formulas were applied to the calculation of the initial magnetic susceptibility. The obtained expression of susceptibility fully coincides with the known theories.
article
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Article history: Received 28 April 2014 Received in revised form 28 July 2014 Available online 10 August 2014 Keywords: Dipolar hard spheres Virial coefficients Diagram technique Free energy Magnetic field Initial magnetic susceptibility
abstract A method for calculation of the free energy of dipolar hard spheres under the presence of an applied magnetic field is presented. The method is based on the virial expansion in terms of density as well as the dipolar coupling constant λ, and it uses diagram technique. The formulas and the diagrams, needed to calculate the second B2 and third B3 virial coefficients, are derived up to the order of ∼ λ3 , and compared to the zero-field case. The formula for B2 is the same as in the zero-field case; the formula for B3 , however, is different in an applied field, and a derivation is presented. This is a surprising result which is not emphasized in standard texts, but which has been noticed before in the virial expansion for flexible molecules (Caracciolo et al., 2006; Caracciolo et al., 2008). To verify the correctness of the obtained formulas, B2 and B3 were calculated within the accuracy of λ2 , which were applied to initial magnetic susceptibility. The obtained expression fully coincides with the well-known theories (Morozov and Lebedev, 1990; Huke and Lücke, 2000; Ivanov and Kuznetsova, 2001), which used different methods to calculate the initial magnetic susceptibility. © 2014 Elsevier B.V. All rights reserved.
1. Introduction The central problem of statistical physics is the study of interacting particle systems. The majority of well-known theories are based on the methods of functional transformations. Thus, for example, in Ref. [1] the collective variable transformation (CV) method was successfully applied to calculate the partition function of Coulomb systems; in Ref. [2], the statistical theory was developed on the basis of the CV method to study the phase separation in ionic fluids. Analytical expressions for pressure and free energy of hard sphere (HS) system with an additional pair isotopic potential were obtained in Ref. [3]
∗
Corresponding author. Tel.: +7 3433507541; fax: +7 3433507541. E-mail addresses: [email protected], [email protected] (E.A. Elfimova).
http://dx.doi.org/10.1016/j.physa.2014.08.002 0378-4371/© 2014 Elsevier B.V. All rights reserved.
E.A. Elfimova et al. / Physica A 415 (2014) 210–219
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using CV and Hubbard–Stratonovich transformation. Another powerful instrument in modern statistical physics to study equilibrium properties of fluids and gases is the method of expansion into series according to the perturbation theory. The classical virial expansion presents the free energy in a series over density [4,5]. Coefficients of this series, known as the virial coefficients, are defined by collective interaction in groups of particles. Caracciolo et al. [6,7] have showed that formulas of the third and fourth virial coefficients are different for the rigid and flexible molecular models because the complex fluids of flexible molecules have some additional intramolecular degrees of freedom. These authors derived the expressions within the context of a polymer solution. This is a surprising result which is not emphasized in standard textbooks. Here, we present our result for the second and third virial coefficients of dipolar hard spheres (DHS) under an applied magnetic field. DHS is traditionally used as a simple model for ferrofluids, which are the stable dispersions of magnetic single-domain nanograins suspended in a carrier liquid [8]. Much of the functionality of ferrofluids arise from the responses of their structure and dynamics to applied magnetic fields. For instance, the diffusion [9], the structural [10,11] and the optical properties [12] become anisotropic under the presence of a uniform magnetic field. Such effects are used in a variety of applications. Anisotropy arises due to the field-induced alignment of particles in the field direction. So, the common feature between flexible molecules and ferrofluids appears to be the presence of intramolecular degrees of freedom contributing to the potential energy, which in the present case corresponds to the dipolar orientation coupling to the applied field. It means, that virial coefficients for DHS fluid in an applied field contain the extra terms, whereas in zero-field and infinite-field cases DHS virial coefficients are determined by standard formulas from textbooks. The main aim of the present paper is to derive the method for calculation of the second and third virial coefficients of the DHS fluid under the presence of an applied magnetic field. This article is arranged as follows. Section 2 contains the definition of the DHS model, the interaction potentials and thermodynamic parameters. Basic concept of diagram method for DHS fluid in zero-field is presented in Section 3. The virial expansion theory for DHS fluid under an applied magnetic field is developed in Section 4. The expression for the second virial coefficient is the same as for the zero-field case; the formula for the third virial coefficient, however, is different under an applied field, and the derivation is presented. In Section 5 we show an outline of the calculations of the second and third virial coefficients within the accuracy of λ2 , which are applied to the initial magnetic susceptibility to verify the correctness of the obtained formulas. The summary is provided in Section 6. 2. Model Let us consider a DHS fluid of N homogeneously magnetized hard spheres of diameter σ confined to a volume V at temperature T . Under the influence of an external magnetic field, the properties of magnetic medium are dependent on the shape of the container due to the demagnetization effect. Therefore, the container is chosen to be a prolate ellipsoid of revolution of infinite elongation aligned along the field direction. In the case of this container shape, the demagnetization factor is zero and the internal magnetic field coincides with the external field H. Each particle has five degrees of freedom: three of them are the translation motion, described with movement of the radius vector ri (ri , θi , φi ) of the ith particle in the system volume, and two degrees of freedom are connected with the rotation of its magnetic moment mi (mi , ωi , ζi ). The external magnetic field H is taken to be parallel with the laboratory z axis. The Hamiltonian of the system contains contributions from the short-range hard sphere repulsion Us (ij), the long-range dipole–dipole potential energy Ud (ij), and the interactions between the dipoles and the external field Um (i)
H = Hd + Hs + Hm ,
Hd =
N
Ud (ij), Hs =
i
Hm =
N
Um (i) = −
i =1
Us (ij) =
Ud (ij) = − 3
Us (ij),
(1)
i
N N (mi , H) = −mH cos ωi , i=1
∞, 0,
N
rij < σ rij > σ ,
i =1
rij = ri − rj ,
(mi , rij )(mj , rij ) rij5
−
(mi , mj ) rij3
.
(2)
The dipole–dipole interaction potential Ud (ij) has a non-central character, that is, it depends on both distance rij between ith and jth particles and the relative orientation of their magnetic moments mi and mj ; moreover, its character can change from attraction to repulsion. The strength of dipolar interaction is measured by the dipolar coupling constant λ = m2 /σ 3 kT , which means the ratio of characteristic interaction energy m2 /σ 3 of magnetic moments of two contacting particles to thermal energy kT . The dipole–field interaction is defined by the Langevin parameter α = mH /kT . The particle concentration is expressed as the volume fraction ϕ = nv , where n = N /V is a number density and v = π σ 3 /6 is a particle volume.
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E.A. Elfimova et al. / Physica A 415 (2014) 210–219 Table 1 (ds)
Diagrams corresponding to the second virial coefficient B2 zero-field DHS fluid. Diagram D1
D2
Coefficient
Formula
1
V 2
ˆ 12 − R
V 2·2!
1
ˆ 12 R
Ud (12) kT
12
(12) − UdkT
up to Ud3 -term for
[fs (12) + 1] 2
[fs (12) + 1] 12
D3
V 2·3!
1
ˆ 12 R
(12) − UdkT
3
[fs (12) + 1] 12
3. Diagrammatic expansion of the DHS free energy at zero-field A classical result of virial expansion [4,5] presents the free energy of hard sphere (HS) fluid as a series in terms of density n F = Fid − NkT
∞
(s)
Bp+1 (λ)np ,
(3)
p=1
(s)
where Fid is the ideal gas contribution and Bp+1 are temperature-dependent virial coefficients. The exact expressions for the second and third virial coefficients have the following form (s)
B2 = (s)
B3 =
V 2
ˆ 1 Rˆ 2 fs (12), R
V2
(4)
ˆ 1 Rˆ 2 Rˆ 3 fs (12)fs (13)fs (23), R
6
(5)
ˆ i denotes an averaging over all where fs (ij) = exp[−Us (ij)/kT ] − 1 is the Mayer function of the HS fluid; the operator R possible positions of the ith particle in volume V ˆ i 1 = V −1 R
1 dri = 1,
dri = ri2 sin θi dri dθi dφi .
(6)
V
Virial expansion for zero-field DHS is derived using the classical virial expansion for HS, described above. The interaction between two particles i, j, according to (1), is defined by a sum of pair potentials U (ij) = Us (ij) + Ud (ij). To calculate virial coefficients it is necessary to average the Mayer functions f (ij) = exp[−Us (ij) − Ud (ij)/kT ] − 1 over both the magnetic moment orientations and the particle positions (ds)
B2
=
⟨1⟩i =
V 2
(ds)
ˆ 1 Rˆ 2 ⟨f (12)⟩12 , R
1 4π
2π
π
0
B3
=
V2 6
ˆ 1 Rˆ 2 Rˆ 3 ⟨f (12)f (13)f (23)⟩123 , R
1 sin ωi dωi dζi = 1.
(7) (8)
0
(ds)
The coefficients Bk
themselves are the functions of dipolar coupling constant λ, which for real ferrofluid is of the order (ds)
of unity. Therefore, following Ref. [13], the virial coefficients Bk
may be expanded in powers of λ
Ud (ij) − . (9) l ! kT l =1 Here the dipolar coupling constant λ plays a part of the ‘‘small parameter’’ of the expansion. In this paper we take into (ds) (ds) (ds) account all terms up to λ3 for B2 and B3 . Using the expansion (9) for the evaluation of Bk (7), we get f (ij) = fs (ij) + [fs (ij) + 1]
(ds)
B2
(ds)
B3
∞ 1
l
= B(2s) + D1 + D2 + D3,
(10)
(s)
= B3 + 3 · D4 + 3 · D5 + 3 · D6 + 3 · D7 + 6 · D8 + D9,
ˆ 1 Rˆ 2 and Rˆ 1 Rˆ 2 Rˆ 3 are replaced by the where the expressions for terms Di are shown in Tables 1 and 2. Integrations over R ˆ 12 and Rˆ 12 Rˆ 13 with particle 1 at the origin. The tables also depict diagram presentation of Di: particles i, j are shown as ones R points; single solid line defines the sum fs (ij)+ 1; double solid line corresponds to fs (ij); one dotted line means [−Ud (ij)/kT ]; two dotted lines are [−Ud (ij)/kT ]2 /2!, and etc. Column ‘‘Coefficient’’ shows the combinatorial factor, which takes into account topologically similar diagrams, obtained due to the permutations of vertex numbers inside the diagram. Calculation of the second and third virial coefficients for zero-field DHS fluid is given in Refs. [13–16]. The above diagrammatic expansion is the method of free energy calculation for both the zero-field DHS fluid and any simple fluid, whose interparticle interaction energy may be presented as a sum of potentials of the basic HS system and pair potential, which is a correction to the basic system.
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Table 2 (ds)
Diagrams corresponding to the third virial coefficient B3 Diagram
D4
Coefficient
Formula
3
V2 6
ˆ 12 Rˆ 13 − R
V2 R R 6·2! 12 13
ˆ
3
D5
up to Ud3 -term for zero-field DHS fluid.
ˆ
Ud (12) kT
12
[fs (12) + 1]fs (13)fs (23)
2
(12) − UdkT
[fs (12) + 1]fs (13)fs (23)
12
V2 R R 6·3! 12 13
ˆ
3
D6
ˆ
3
(12) − UdkT
[fs (12) + 1]fs (13)fs (23)
12
D7
3
V2 6
D8
6
V2 R R 6·2! 12 13
D9
1
V2 6
ˆ 12 Rˆ 13 R
ˆ
ˆ
ˆ 12 Rˆ 13 R
[−Ud (12)][−Ud (13)] (kT )2 123
[fs (12) + 1][fs (13) + 1]fs (23)
[−Ud (12)]2 [−Ud (13)] (kT )3 123
[fs (12) + 1][fs (13) + 1]fs (23)
[−Ud (12)][−Ud (13)][−Ud (23)] (kT )3 123
[fs (12)+ 1][fs (13)+ 1][fs (23)+ 1]
4. Diagrammatic expansion of DHS free energy under an applied magnetic field When a uniform magnetic field is applied, the particle magnetic moments of DHSs interact with each other and with the field. This leads to appearance of extra degrees of freedom due to the dipolar orientation coupling to the applied field. It significantly influences the structure of classical free energy virial expansion. A full derivation is given here, the structure of which follows closely Section 6 of Ref. [4]. Helmholtz free energy of DHS fluid under an applied magnetic field is
F = Fid + Fm − kT ln Q ,
Fm = −NkT ln
sinh α
α
,
(11)
where Fm is the free energy of ideal paramagnetic gas, Q stands for the configuration part of the partition function. Indicating the HS partition function as Qs , the third term on the right-hand side of (11) may be presented as follows
−kT ln Q = −kT ln Qs + △F ,
△F = −kT ln
Q Qs
.
(12)
Using the definition of the DHS partition function under an applied field, the △F can be written as
N ˆs ˆd H H 1 ˆ △F = −kT ln = −kT ln Ri exp − − Qs Qs i=1 kT kT
Q
N
ˆi
,
(13)
i=1
where ⟨· · ·⟩ˆ i is the averaging over all possible orientations of the ith magnetic moment with account for the dipole–field interaction
⟨1⟩ˆ i =
1
α
4π sinh α
ˆ d /kT In Eq. (13), exp −H
2π
0
π
1 · exp (α cos ωi ) sin ωi dωi dζi = 1.
(14)
0
is the expanded Taylor series
N
ˆ 1 ˆ i exp − Hs △F = −kT ln R Qs i=1
kT
1+
∞ p=1
1 p!
−
ˆd H kT
p N i=1
ˆi
.
(15)
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The first terms of the series (15) provide a major contribution to △F when the interparticle dipole–dipole interaction intensity is comparable with the thermal energy. In fact, in the general case, the number of terms under consideration depends on the type and intensity of interparticle interaction. Here the first, the second and the third term of the series are presented. ˆ i Rˆ j Ud (ij) = As all particles of the DHS model are identical, the contribution from particles i, j and 1, 2 into △F is similar: R
ˆ 1 Rˆ 2 Ud (12). Therefore, combinatorial counting of these terms is necessary [4]. R − 1
ˆd H kT
=
−
2!
N 1
kT i,j=1
ˆd H
−
2
2!
N (N − 1)
⇒
kT 1
=
kT
Ud (ij)
N
2
−
Ud (12) − ,
(16)
kT
2
Ud (ij)
=
1
2 N Ud (ij) −
kT 2! i
i ,j = 1 N
+
kT
i̸=j̸=m=1
N (N − 1)
⇒
+ 1
3!
−
ˆd H
3
kT
−
+
+
(kT )2
−
Ud (12)
3 +
kT
Ud (ij) kT
Ud (mp) − kT
N (N − 1)(N − 2) [−Ud (12)][−Ud (13)]
2!4
3!2
+
Ud (12)
2
−
i̸=j̸=m̸=p=1
2!2 kT 2! N (N − 1)(N − 2)(N − 3) [−Ud (12)][−Ud (34)]
N (N − 1)
⇒
kT
(kT )2 ,
(17)
3N (N − 1)(N − 2) [−Ud (12)]2 [−Ud (13)]
(kT )3
3!
N (N − 1)(N − 2) [−Ud (12)][−Ud (13)][−Ud (23)] 3! (kT )3 3N (N − 1)(N − 2)(N − 3) [−Ud (12)][−Ud (34)]2
(kT )3
3!4
+ ···.
(18)
ˆ d /kT )3 is presented up to the fourth order in terms of the particle number N 4 . Each factor of Above, the Hamiltonian (H −Ud (ij)/kT yields an additional power of the dipolar coupling constant λ. Using the expressions (16)–(18) for evaluation of △F (15) up to the order λ3 N 4 , and collecting together all terms of equal order in λ, we get △F = −kT ln 1 + A(λ) + B(λ2 ) + C (λ3 ) .
(19)
Here, the terms in the logarithm argument are clearly defined as functions of λ. However, these functions also depend on the density n, the number of particles N and the system volume V . The exact definitions of the terms are presented in Appendix A. Performing cumulant expansion (Section 6 of Ref. [4]) of the logarithm (19) within the accuracy of λ3 , we obtain
A2 (λ) 1 △F = −kT A(λ) + B(λ2 ) + C (λ3 ) − − A(λ)B(λ2 ) + A3 (λ) + · · · . 2
(20)
3
Since each term (20) is density dependent, we may regroup △F into series over n collecting terms with equal order of n
△F = −NkT
∞
(fds)
Bk+1 (λ)nk .
(21)
k=1
(fds)
(fds)
The regrouping procedure is derived in Appendix B for coefficients B2 (fds)
classical virial series (3), and the coefficients Bi (fds)
B3
and B3
in detail. Series (21) is analogous to the
(λ) are similar to the virial coefficients [4,5]. Coefficients B(2fds) (λ) and
(λ) up to terms of order of λ3 are given by (fds)
B2
(fds)
B3
= D1(f ) + D2(f ) + D3(f ) , (f )
= 3 · D4
(f )
+ 3 · D5
(22) (f )
+ 3 · D6
(f )
+ 3 · D7
+ 6 · D8
(f )
(f )
+ D9
− D10
(f )
− 2 · D11
(f )
(23)
where formulas for Di(f ) are shown in Tables 3 and 4. The expression and the diagram presentation for the second virial coefficient under an applied field is the same as for the zero-field case. The difference is only in the averaging operators ⟨· · ·⟩ˆ i
⟨· · ·⟩i . The structure of the third virial coefficient under an applied field B(3fds) differs from the case of zero field: unlike B(3ds) , (fds)
the coefficient B3 contains two additional diagrams D10(f ) and D11(f ) ; also the structure of diagrams D7(f ) and D8(f ) differs from that of D7 and D8. Nevertheless, (23) is the complete formula for the third virial coefficient within the accuracy of λ3 , and covers zero-, finite- and infinite-field cases. In zero-field (α = 0) or in an infinite field (α = ∞), 3 · D7(f ) − D10(f ) = D7 and 6 · D8(f ) − 2 · D11(f ) = D8. The proof is below.
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Table 3 (fds)
Diagrams corresponding to the second virial coefficient B2 DHS fluid under an applied field. Diagram (f )
D1
D2(f )
Coefficient
Formula
1
V 2
1
D3(f )
1
ˆ 12 − R
Ud (12) kT
up to the Ud3 -term for
ˆ 1 ˆ2
[fs (12) + 1]
V 2·2!
2 ˆ 12 − Ud (12) R kT
[fs (12) + 1]
V 2·3!
3 ˆ 12 − Ud (12) R kT
[fs (12) + 1]
ˆ2 ˆ 1
ˆ 1 ˆ2
Table 4 (fds)
up to the Ud3 -term for DHS fluid in an applied field.
Diagrams corresponding to the third virial coefficient B3 Diagram
Coefficient
Formula
D4(f )
3
V2 6
D5(f )
3
V2 R R 6·2! 12 13
D6(f )
3
ˆ 12 Rˆ 13 − R
ˆ
ˆ
V2 R R 6·3! 12 13
ˆ
ˆ
Ud (12) kT
ˆ 1 ˆ2
(12) − UdkT
2
(12) − UdkT
3
[fs (12) + 1]fs (13)fs (23)
[fs (12) + 1]fs (13)fs (23) ˆ 1 ˆ2
[fs (12) + 1]fs (13)fs (23) ˆ 1 ˆ2
[−Ud (12)][−Ud (13)] (kT )2 ˆ 1 ˆ 2 ˆ3
D7(f )
3
V2 6
D8(f )
6
V2 R R 6·2! 12 13
D9(f )
1
V2 6
ˆ 12 Rˆ 13 R
[−Ud (12)][−Ud (13)][−Ud (23)] (kT )3 ˆ 1 ˆ 2 ˆ3
D10(f )
1
V2 2
ˆ 12 Rˆ 34 R
[−Ud (12)][−Ud (34)] (kT )2 ˆ 1 ˆ 2 ˆ 3 ˆ4
D11(f )
2
V2 R R 2·2! 12 34
ˆ 12 Rˆ 13 R
ˆ
ˆ
ˆ
ˆ
[fs (12) + 1][fs (13) + 1][fs (23) + 1]
[−Ud (12)]2 [−Ud (13)] (kT )3 ˆ 1 ˆ 2 ˆ3
[fs (12) + 1][fs (13) + 1][fs (23) + 1]
[fs (12) + 1][fs (13) + 1][fs (23) + 1]
[fs (12) + 1][fs (34) + 1]
[−Ud (12)]2 [−Ud (34)] (kT )3 ˆ 1 ˆ 2 ˆ 3 ˆ4
[fs (12) + 1][fs (34) + 1]
Diagram D7(f ) may be presented as follows D7(f ) =
[−Ud (12)][−Ud (13)] [fs (12) + 1][fs (13) + 1]fs (23) 6 (kT )2 ˆ 1 ˆ 2 ˆ3 V2 [−Ud (12)][−Ud (13)] ˆ ˆ + R12 R13 [fs (12) + 1][fs (13) + 1]. 6 (kT )2 ˆ 1 ˆ 2 ˆ3 V2
ˆ 12 Rˆ 13 R
(24)
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The first term (24) corresponds exactly to the diagram D7. The second term (24) will be completely compensated by the diagram D10(f ) in zero- and infinite-field, because in these cases
[−Ud (12)][−Ud (13)] (kT )2
Ud (12) Ud (34) ≡ − − .
kT
123
kT
12
34
Under the finite applied field, the averaging over magnetic moment orientation depends on the angle between the radius vector of the particle position and the field direction, therefore
[−Ud (12)][−Ud (13)] (kT )2
̸= −
Ud (12) kT
ˆ3 ˆ 2 ˆ 1
−
Ud (34) kT
ˆ2 ˆ 1
, ˆ4 ˆ 3
so the second term (24) and the diagram D10(f ) give non-zero contributions to the third virial coefficient. Similar discussion may be applied to the diagrams D8(f ) and D11(f ) . The main conclusion from this section is the following: the expression for the DHS second virial coefficient under an applied field is the same as in the zero- and infinite-field cases; in the finite fields, the structure of the DHS third virial coefficient differs from the case of zero- and infinite-fields. The exact expression for the third virial coefficient within the accuracy of λ3 , covering zero-, finite-, and infinite-field cases, is given by (23). 5. Initial magnetic susceptibility According to the known models [17–19], the initial magnetic susceptibility of the DHS fluid is well described by the following expression
4π χL (4π χL )2 χ = χL 1 + + , 3
(25)
144
where χL = 2ϕλ/π is the Langevin susceptibility. This expression, obtained with different methods, has been proved as giving excellent agreement with the computer simulations both for the ferrofluids [20] and the ferrogels [21]. (fds) (fds) To verify the correctness of the formulas for B2 and B3 , the initial magnetic susceptibility is calculated using the free energy virial expansion
1 m 2 ∂ 2 F , χ = lim − α→0 V kT ∂α 2
(26)
(fds)
F = Fid + Fm − kT ln Qs − NkT B2
(fds) 2
n + B3
n
.
The terms Fid and kT ln Qs do not depend on the Langevin parameter α , therefore they do not contribute to χ . The contribution from the free energy of the ideal paramagnetic gas Fm to χ is equal to χL . It is the first term in (25). (fds) (fds) ˆ j . The results of the To calculate B2 (22) and B3 (23) it is necessary to perform the integration over ⟨· · ·⟩ˆ i and R
ˆ j is described in orientational averagings ⟨· · ·⟩ˆ i are given in Ref. [22]. A procedure of averaging over particle positions R (fds)
Refs. [22,23] in detail. The explicit expressions for B2 (fds)
B2
= D1(f ) + D2(f ) = 4vλL2 (α) +
4v 3
(fds)
and B3
within the accuracy of λ2 are as follows
L2 (α) λ2 1 + 3 , 5
(fds)
= 3 · D4(f ) + 3 · D5(f ) + 3 · D7(f ) − D10(f ) , 1 L23 (α) (f ) (f ) 2 2 D4 = 0, D5 = v λ 2 ln 2 + 1+ 3 5 L(α) L(α) D7(f ) = 2v 2 λ2 L2 (α) 3 − 6 + 2v 2 λ2 L2 (α) 16 − 32 α α B3
D10(f ) = 32v 2 λ2 L4 (α) L(α) = coth α − 1/α,
L3 (α) = 1 − (fds)
3L(α)
α
(27) (28)
.
The contribution from −NkTB2 to χ is equal to 4π χL2 /3. It coincides with the second term in (25). The result for D7(f ) is divided into two terms, similar to (24), where the second term together with D10(f ) is an extra contribution to the third virial coefficient, conditioned by an applied magnetic field. It is easy to see that their sum gives a (fds) non-zero contribution to χ , and this is equal to χL (4π χL )2 /9. Finally, the total contribution from −NkTB3 to χ is equal to χL (4π χL )2 /144. It coincides with the third term in (25).
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6. Conclusion Expressions for the second and third coefficients of DHS free energy virial expansion are derived for the case of an applied magnetic field of various strength. Virial coefficients are presented in a form of series over the dipolar coupling constant λ. The formulas obtained have the accuracy of λ3 . Truncation at this level is accurate for physically realistic parameters λ ≃ 1. A diagram interpretation of the virial coefficients is offered. The DHS second virial coefficient has the same structure in zero-, finite-, and infinite-fields. The field-dependent DHS third virial coefficient differs from the cases of zero- and infinitefields, and the derivation is presented. Complete formula for the third virial coefficient within the accuracy of λ3 , covering zero-, finite-, and infinite-fields, is given. In comparison with normally-quoted result for the third virial coefficient of simple fluids the complete formula takes into account additional contribution, which compensates for each other in the zero- and infinite-field cases, but are not equal to zero in the finite-field case. The field-dependent second and third virial coefficients are calculated within the accuracy of λ2 . They are applied to the initial magnetic susceptibility of a DHS fluid. The correct result for the initial magnetic susceptibility, coinciding with those of Refs. [17–19], can be obtained only under the condition that the additional contribution in the third virial coefficient is taken into account. Acknowledgement The research is supported by the Ministry of Education and Science of the Russian Federation grant N 3.12.2014/K and contract N 02.A03.21.0006 from 27.03.2013. Appendix A
A(λ) =
=
N (N − 1) 2 N (N − 1) 2
ˆ 1 Rˆ 2 − R
ˆ 1 Rˆ 2 − R
Ud (12) kT Ud (12)
N 1
ˆ 1 ˆ2
kT
Qs i=3
ˆ i exp − R
ˆs H
kT
(s)
ˆ 1 ˆ2
g2 (r ),
(A.1)
B(λ2 ) = B1(λ2 ) + B2(λ2 ) + B3(λ2 ), B1(λ2 ) =
=
N (N − 1) 2!2 N (N − 1) 4
ˆ 1 Rˆ 2 R ˆ 1 Rˆ 2 R
− −
(A.2)
Ud (12)
2
kT
1
ˆ 1 ˆ2
Ud (12)
2
kT
N
Qs i=3
ˆ i exp − R
ˆs H
kT
(s)
g2 (r ),
ˆ 1 ˆ2
N ˆs H [− Ud (12)][−Ud (13)] 1 ˆ 1 Rˆ 2 Rˆ 3 ˆ i exp − R R B2(λ ) = 2! (kT )2 kT ˆ ˆ ˆ Qs i=4 1 2 3 N (N − 1)(N − 2) [− Ud (12)][−Ud (13)] (s) ˆ 1 Rˆ 2 Rˆ 3 R = g3 (r ), 2 (kT )2 ˆ 1 ˆ 2 ˆ3 N ˆs [− Ud (12)][−Ud (34)] 1 H N (N − 1)(N − 2)(N − 3) 2 ˆ 1 Rˆ 2 Rˆ 3 Rˆ 4 ˆ i exp − R R B3(λ ) = 2!4 (kT )2 kT ˆ 1 ˆ 2 ˆ 3 ˆ 4 Qs i=5 [− Ud (12)][−Ud (34)] N (N − 1)(N − 2)(N − 3) (s) ˆ 1 Rˆ 2 Rˆ 3 Rˆ 4 = R g4 (r ), 8 (kT )2 ˆ 1 ˆ 2 ˆ 3 ˆ4 N (N − 1)(N − 2)
2
C (λ3 ) = C 1(λ3 ) + C 2(λ3 ) + C 3(λ3 ) + C 4(λ3 ) C 1(λ3 ) =
=
C 2(λ3 ) =
N (N − 1) 3!2 N (N − 1) 12
ˆ 1 Rˆ 2 R ˆ 1 Rˆ 2 R
− −
3N (N − 1)(N − 2)
Ud (12)
3
kT
ˆ 1 ˆ2
Ud (12)
3
kT
ˆ 1 Rˆ 2 Rˆ 3 R
(A.3)
N ˆ 1 ˆRi exp − Hs
Qs i=3
kT
(s)
g2 (r ),
ˆ 1 ˆ2
[−Ud (12)]2 [−Ud (13)] (kT )3
3! ˆ ˆ ˆ 1 2 3 N 2 ˆ 1 (s) ˆ i exp − Hs N (N − 1)(N − 2) Rˆ 1 Rˆ 2 Rˆ 3 [−Ud (12)] [−Ud (13)] × R g3 (r ), Qs i=4 kT 2 (kT )3 ˆ ˆ ˆ 1 2 3
218
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N ˆs [−Ud (12)][−Ud (13)][−Ud (23)] 1 H ˆ ˆ ˆ ˆ Ri exp − C 3(λ ) = R1 R2 R3 3! (kT )3 kT ˆ ˆ ˆ Qs i=4 1 2 3 N (N − 1)(N − 2) (s) ˆ 1 Rˆ 2 Rˆ 3 [−Ud (12)][−Ud (13)][−Ud (23)] g3 (r ), R = 6 (kT )3 ˆ ˆ ˆ 1 2 3 N 2 ˆs 3N (N − 1)(N − 2)(N − 3) [−Ud (12)][−Ud (34)] 1 H 3 ˆ ˆ ˆ ˆ ˆ C 4(λ ) = Ri exp − R1 R2 R3 R4 3!4 (kT )3 kT ˆ 4 Qs i=5 ˆ 3 ˆ 2 ˆ 1 [− Ud (12)][−Ud (34)]2 N (N − 1)(N − 2)(N − 3) (s) ˆ 1 Rˆ 2 Rˆ 3 Rˆ 4 g4 (r ), R = 8 (kT )3 ˆ4 ˆ 3 ˆ 2 ˆ 1 N ˆs 1 H (s) ˆ gk = Ri exp − . N (N − 1)(N − 2)
3
Qs i=k+1
kT
(A.4)
Appendix B (s)
Notice, the terms A(λ), B(λ2 ) and C (λ3 ) contain a partial distribution function of HS fluid gi , which can be presented as a series in terms of density n [4] (s)
ˆ 3 fs (13)fs (23)) + O(n2 )], g2 (r ) = (fs (12) + 1)[1 + nV (R
(B.5)
(s)
g3 (r ) = (fs (12) + 1)(fs (13) + 1)(fs (23) + 1) + O(n)
(B.6)
(s)
g4 (r ) = (fs (12) + 1)(fs (13) + 1)(fs (14) + 1)(fs (23) + 1)(fs (24) + 1)(fs (34) + 1) + O(n).
(B.7)
ˆ j , (j ̸= 1) may be replaced by Rˆ 1j with particle 1 at the origin; in this case averaging Rˆ 1 yields 1. Averaging R • Coefficient B(2fds) (λ). (s)
In the lowest order of density in g2 (r ), the terms A(λ), B1(λ2 ) and C 1(λ3 ) have the orders of Nnλ, Nnλ2 and Nnλ3 , respectively. Here, the approximation of the thermodynamic limit (N − 1)/V = n is used. The remaining terms of (20) have (fds) in thermodynamic limit the orders higher than λ3 or higher than n. Thus, B2 (λ) has a form (fds)
B2
(λ) =
V 2
2 3 ˆR12 − Ud (12) + 1 − Ud (12) + 1 − Ud (12) kT
2
kT
6
kT
[fs (12) + 1],
(B.8)
ˆ 1 ˆ2
ˆ 1 Rˆ 2 is replaced by one Rˆ 12 with particle 1 at the origin. where integration over R • Coefficient B(3fds) (λ). (s)
In the first order of density in g2 (r ) the same terms A(λ), B1(λ2 ), and C 1(λ3 ) are proportional to n2 in thermodynamic (fds)
limit. Thus, they yield contribution to the coefficient B3
ˆ 12 Rˆ 13 − A(λ) + B1(λ ) + C 1(λ ) ⇒ Nn V R 2
3
2
2
(λ)
1 Ud (12) 2
kT
+
1 4
−
Ud (12) kT
2 +
1 12
−
Ud (12) kT
3 ˆ 1 ˆ2
× [fs (12) + 1]fs (13)fs (23). (s)
(B.9)
In the lowest order of density in g3 (r ), terms B2(λ ), C 2(λ ), and C 3(λ ) have the orders of Nn λ , Nn λ , and Nn2 λ3 , respectively, in approximation of the thermodynamic limit (N − 1)(N − 2)/V 2 = n2 . Thus, they also yield a contribution to (fds) the coefficient B3 (λ) 2
3
3
ˆ 12 Rˆ 13 [fs (12) + 1][fs (13) + 1][fs (23) + 1] B2(λ2 ) + C 2(λ3 ) + C 3(λ3 ) ⇒ Nn2 V 2 R
2 2
2 3
1 [−Ud (12)][−Ud (13)]
(kT )2 1 [−Ud2 (12)][−Ud (13)] 1 [−Ud (12)][−Ud (13)][−Ud (23)] . + + 2 (kT )3 6 (kT )3 ˆ 1 ˆ 2 ˆ3 2
(B.10)
Terms B3(λ2 ) and A2 (λ)/2 should be considered together, because in the thermodynamic limit and in the lowest order (s) (s) of density in g4 (r ) and g2 (r ) each term has an order of N 2 n2 λ2 , which indicates the divergency ∼ N 2 ; the sum of them,
E.A. Elfimova et al. / Physica A 415 (2014) 210–219
219
however, has the contribution of the order of Nn2 λ2 B3(λ ) − A (λ)/2 ⇒ 2
2
N (N − 1)(N − 2)(N − 3) − N 2 (N − 1)2
8
[−Ud (12)][−Ud (34)] [fs (12) + 1][fs (34) + 1] (kT )2 ˆ4 ˆ 3 ˆ 2 ˆ 1 V 2 Nn2 ˆ 12 Rˆ 34 [−Ud (12)][−Ud (34)] ⇒− [fs (12) + 1][fs (34) + 1], R 2 (kT )2 ˆ4 ˆ 3 ˆ 2 ˆ 1
× Rˆ 12 Rˆ 34
(B.11)
ˆ 12 and Rˆ 34 are performed independently in different frames with particles 1 and 3 at the origin, here, the averagings R respectively. Similarly, the terms C 4(λ3 ) and A(λ)B1(λ2 ) have divergent of ∼ N 2 separately, whereas their sum in the thermodynamic (s) (s) limit and in the lower order of partial distribution functions g4 (r ) and g2 (r ) defines the term of order Nn2 λ3 C 4(λ ) − A(λ)B1(λ ) ⇒ − 2
2
V 2 Nn2 2
2 ˆR12 Rˆ 34 [fs (12) + 1][fs (34) + 1] [−Ud (12)][−Ud (34)] . (kT )3 ˆ 1 ˆ 2 ˆ 3 ˆ4
(B.12)
In the thermodynamic limit approximation the remaining terms of (20) have the orders higher than n2 or higher than λ3 . (fds) Collecting terms (B.9)–(B.12), the coefficient B3 (λ) has a form within the accuracy of λ3 (fds)
B3
(λ) = V Rˆ 12 2
+ Rˆ 13
ˆ 13 − R
1 Ud (12) 2
kT
+
1
4
1 [−Ud (12)][−Ud (13)]
−
+
Ud (12) kT
2 +
1 12
−
Ud (12) kT
3
[fs (12) + 1]fs (13)fs (23) ˆ 1 ˆ2
1 [−Ud2 (12)][−Ud (13)]
(kT )2 2 (kT )3 1 [−Ud (12)][−Ud (13)][−Ud (23)] + [fs (12) + 1][fs (13) + 1][fs (23) + 1] 6 (kT )3 ˆ 1 ˆ 2 ˆ3 [−Ud2 (12)][−Ud (34)] 1 [−Ud (12)][−Ud (34)] ˆ + − R34 [fs (12) + 1][fs (34) + 1] . 2 (kT )2 (kT )3 ˆ 1 ˆ 2 ˆ 3 ˆ4 2
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(B.13)
Contents lists available at ScienceDirect
Physica A journal homepage: www.elsevier.com/locate/physa
Free energy of dipolar hard spheres: The virial expansion under the presence of an external magnetic field Ekaterina A. Elfimova ∗ , Tatyana E. Karavaeva, Alexey O. Ivanov Institute of Mathematics and Computer Sciences, Ural Federal University, 51 Lenin Avenue, Ekaterinburg 620000, Russia
highlights • • • • •
The model of dipolar hard spheres in an external magnetic field was considered. The expressions for 2nd (B2) and 3rd (B3) virial coefficients were derived. The formula for B3 for a system in the field is different from the zero-field case. The formulas were applied to the calculation of the initial magnetic susceptibility. The obtained expression of susceptibility fully coincides with the known theories.
article
info
Article history: Received 28 April 2014 Received in revised form 28 July 2014 Available online 10 August 2014 Keywords: Dipolar hard spheres Virial coefficients Diagram technique Free energy Magnetic field Initial magnetic susceptibility
abstract A method for calculation of the free energy of dipolar hard spheres under the presence of an applied magnetic field is presented. The method is based on the virial expansion in terms of density as well as the dipolar coupling constant λ, and it uses diagram technique. The formulas and the diagrams, needed to calculate the second B2 and third B3 virial coefficients, are derived up to the order of ∼ λ3 , and compared to the zero-field case. The formula for B2 is the same as in the zero-field case; the formula for B3 , however, is different in an applied field, and a derivation is presented. This is a surprising result which is not emphasized in standard texts, but which has been noticed before in the virial expansion for flexible molecules (Caracciolo et al., 2006; Caracciolo et al., 2008). To verify the correctness of the obtained formulas, B2 and B3 were calculated within the accuracy of λ2 , which were applied to initial magnetic susceptibility. The obtained expression fully coincides with the well-known theories (Morozov and Lebedev, 1990; Huke and Lücke, 2000; Ivanov and Kuznetsova, 2001), which used different methods to calculate the initial magnetic susceptibility. © 2014 Elsevier B.V. All rights reserved.
1. Introduction The central problem of statistical physics is the study of interacting particle systems. The majority of well-known theories are based on the methods of functional transformations. Thus, for example, in Ref. [1] the collective variable transformation (CV) method was successfully applied to calculate the partition function of Coulomb systems; in Ref. [2], the statistical theory was developed on the basis of the CV method to study the phase separation in ionic fluids. Analytical expressions for pressure and free energy of hard sphere (HS) system with an additional pair isotopic potential were obtained in Ref. [3]
∗
Corresponding author. Tel.: +7 3433507541; fax: +7 3433507541. E-mail addresses: [email protected], [email protected] (E.A. Elfimova).
http://dx.doi.org/10.1016/j.physa.2014.08.002 0378-4371/© 2014 Elsevier B.V. All rights reserved.
E.A. Elfimova et al. / Physica A 415 (2014) 210–219
211
using CV and Hubbard–Stratonovich transformation. Another powerful instrument in modern statistical physics to study equilibrium properties of fluids and gases is the method of expansion into series according to the perturbation theory. The classical virial expansion presents the free energy in a series over density [4,5]. Coefficients of this series, known as the virial coefficients, are defined by collective interaction in groups of particles. Caracciolo et al. [6,7] have showed that formulas of the third and fourth virial coefficients are different for the rigid and flexible molecular models because the complex fluids of flexible molecules have some additional intramolecular degrees of freedom. These authors derived the expressions within the context of a polymer solution. This is a surprising result which is not emphasized in standard textbooks. Here, we present our result for the second and third virial coefficients of dipolar hard spheres (DHS) under an applied magnetic field. DHS is traditionally used as a simple model for ferrofluids, which are the stable dispersions of magnetic single-domain nanograins suspended in a carrier liquid [8]. Much of the functionality of ferrofluids arise from the responses of their structure and dynamics to applied magnetic fields. For instance, the diffusion [9], the structural [10,11] and the optical properties [12] become anisotropic under the presence of a uniform magnetic field. Such effects are used in a variety of applications. Anisotropy arises due to the field-induced alignment of particles in the field direction. So, the common feature between flexible molecules and ferrofluids appears to be the presence of intramolecular degrees of freedom contributing to the potential energy, which in the present case corresponds to the dipolar orientation coupling to the applied field. It means, that virial coefficients for DHS fluid in an applied field contain the extra terms, whereas in zero-field and infinite-field cases DHS virial coefficients are determined by standard formulas from textbooks. The main aim of the present paper is to derive the method for calculation of the second and third virial coefficients of the DHS fluid under the presence of an applied magnetic field. This article is arranged as follows. Section 2 contains the definition of the DHS model, the interaction potentials and thermodynamic parameters. Basic concept of diagram method for DHS fluid in zero-field is presented in Section 3. The virial expansion theory for DHS fluid under an applied magnetic field is developed in Section 4. The expression for the second virial coefficient is the same as for the zero-field case; the formula for the third virial coefficient, however, is different under an applied field, and the derivation is presented. In Section 5 we show an outline of the calculations of the second and third virial coefficients within the accuracy of λ2 , which are applied to the initial magnetic susceptibility to verify the correctness of the obtained formulas. The summary is provided in Section 6. 2. Model Let us consider a DHS fluid of N homogeneously magnetized hard spheres of diameter σ confined to a volume V at temperature T . Under the influence of an external magnetic field, the properties of magnetic medium are dependent on the shape of the container due to the demagnetization effect. Therefore, the container is chosen to be a prolate ellipsoid of revolution of infinite elongation aligned along the field direction. In the case of this container shape, the demagnetization factor is zero and the internal magnetic field coincides with the external field H. Each particle has five degrees of freedom: three of them are the translation motion, described with movement of the radius vector ri (ri , θi , φi ) of the ith particle in the system volume, and two degrees of freedom are connected with the rotation of its magnetic moment mi (mi , ωi , ζi ). The external magnetic field H is taken to be parallel with the laboratory z axis. The Hamiltonian of the system contains contributions from the short-range hard sphere repulsion Us (ij), the long-range dipole–dipole potential energy Ud (ij), and the interactions between the dipoles and the external field Um (i)
H = Hd + Hs + Hm ,
Hd =
N
Ud (ij), Hs =
i
Hm =
N
Um (i) = −
i =1
Us (ij) =
Ud (ij) = − 3
Us (ij),
(1)
i
N N (mi , H) = −mH cos ωi , i=1
∞, 0,
N
rij < σ rij > σ ,
i =1
rij = ri − rj ,
(mi , rij )(mj , rij ) rij5
−
(mi , mj ) rij3
.
(2)
The dipole–dipole interaction potential Ud (ij) has a non-central character, that is, it depends on both distance rij between ith and jth particles and the relative orientation of their magnetic moments mi and mj ; moreover, its character can change from attraction to repulsion. The strength of dipolar interaction is measured by the dipolar coupling constant λ = m2 /σ 3 kT , which means the ratio of characteristic interaction energy m2 /σ 3 of magnetic moments of two contacting particles to thermal energy kT . The dipole–field interaction is defined by the Langevin parameter α = mH /kT . The particle concentration is expressed as the volume fraction ϕ = nv , where n = N /V is a number density and v = π σ 3 /6 is a particle volume.
212
E.A. Elfimova et al. / Physica A 415 (2014) 210–219 Table 1 (ds)
Diagrams corresponding to the second virial coefficient B2 zero-field DHS fluid. Diagram D1
D2
Coefficient
Formula
1
V 2
ˆ 12 − R
V 2·2!
1
ˆ 12 R
Ud (12) kT
12
(12) − UdkT
up to Ud3 -term for
[fs (12) + 1] 2
[fs (12) + 1] 12
D3
V 2·3!
1
ˆ 12 R
(12) − UdkT
3
[fs (12) + 1] 12
3. Diagrammatic expansion of the DHS free energy at zero-field A classical result of virial expansion [4,5] presents the free energy of hard sphere (HS) fluid as a series in terms of density n F = Fid − NkT
∞
(s)
Bp+1 (λ)np ,
(3)
p=1
(s)
where Fid is the ideal gas contribution and Bp+1 are temperature-dependent virial coefficients. The exact expressions for the second and third virial coefficients have the following form (s)
B2 = (s)
B3 =
V 2
ˆ 1 Rˆ 2 fs (12), R
V2
(4)
ˆ 1 Rˆ 2 Rˆ 3 fs (12)fs (13)fs (23), R
6
(5)
ˆ i denotes an averaging over all where fs (ij) = exp[−Us (ij)/kT ] − 1 is the Mayer function of the HS fluid; the operator R possible positions of the ith particle in volume V ˆ i 1 = V −1 R
1 dri = 1,
dri = ri2 sin θi dri dθi dφi .
(6)
V
Virial expansion for zero-field DHS is derived using the classical virial expansion for HS, described above. The interaction between two particles i, j, according to (1), is defined by a sum of pair potentials U (ij) = Us (ij) + Ud (ij). To calculate virial coefficients it is necessary to average the Mayer functions f (ij) = exp[−Us (ij) − Ud (ij)/kT ] − 1 over both the magnetic moment orientations and the particle positions (ds)
B2
=
⟨1⟩i =
V 2
(ds)
ˆ 1 Rˆ 2 ⟨f (12)⟩12 , R
1 4π
2π
π
0
B3
=
V2 6
ˆ 1 Rˆ 2 Rˆ 3 ⟨f (12)f (13)f (23)⟩123 , R
1 sin ωi dωi dζi = 1.
(7) (8)
0
(ds)
The coefficients Bk
themselves are the functions of dipolar coupling constant λ, which for real ferrofluid is of the order (ds)
of unity. Therefore, following Ref. [13], the virial coefficients Bk
may be expanded in powers of λ
Ud (ij) − . (9) l ! kT l =1 Here the dipolar coupling constant λ plays a part of the ‘‘small parameter’’ of the expansion. In this paper we take into (ds) (ds) (ds) account all terms up to λ3 for B2 and B3 . Using the expansion (9) for the evaluation of Bk (7), we get f (ij) = fs (ij) + [fs (ij) + 1]
(ds)
B2
(ds)
B3
∞ 1
l
= B(2s) + D1 + D2 + D3,
(10)
(s)
= B3 + 3 · D4 + 3 · D5 + 3 · D6 + 3 · D7 + 6 · D8 + D9,
ˆ 1 Rˆ 2 and Rˆ 1 Rˆ 2 Rˆ 3 are replaced by the where the expressions for terms Di are shown in Tables 1 and 2. Integrations over R ˆ 12 and Rˆ 12 Rˆ 13 with particle 1 at the origin. The tables also depict diagram presentation of Di: particles i, j are shown as ones R points; single solid line defines the sum fs (ij)+ 1; double solid line corresponds to fs (ij); one dotted line means [−Ud (ij)/kT ]; two dotted lines are [−Ud (ij)/kT ]2 /2!, and etc. Column ‘‘Coefficient’’ shows the combinatorial factor, which takes into account topologically similar diagrams, obtained due to the permutations of vertex numbers inside the diagram. Calculation of the second and third virial coefficients for zero-field DHS fluid is given in Refs. [13–16]. The above diagrammatic expansion is the method of free energy calculation for both the zero-field DHS fluid and any simple fluid, whose interparticle interaction energy may be presented as a sum of potentials of the basic HS system and pair potential, which is a correction to the basic system.
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213
Table 2 (ds)
Diagrams corresponding to the third virial coefficient B3 Diagram
D4
Coefficient
Formula
3
V2 6
ˆ 12 Rˆ 13 − R
V2 R R 6·2! 12 13
ˆ
3
D5
up to Ud3 -term for zero-field DHS fluid.
ˆ
Ud (12) kT
12
[fs (12) + 1]fs (13)fs (23)
2
(12) − UdkT
[fs (12) + 1]fs (13)fs (23)
12
V2 R R 6·3! 12 13
ˆ
3
D6
ˆ
3
(12) − UdkT
[fs (12) + 1]fs (13)fs (23)
12
D7
3
V2 6
D8
6
V2 R R 6·2! 12 13
D9
1
V2 6
ˆ 12 Rˆ 13 R
ˆ
ˆ
ˆ 12 Rˆ 13 R
[−Ud (12)][−Ud (13)] (kT )2 123
[fs (12) + 1][fs (13) + 1]fs (23)
[−Ud (12)]2 [−Ud (13)] (kT )3 123
[fs (12) + 1][fs (13) + 1]fs (23)
[−Ud (12)][−Ud (13)][−Ud (23)] (kT )3 123
[fs (12)+ 1][fs (13)+ 1][fs (23)+ 1]
4. Diagrammatic expansion of DHS free energy under an applied magnetic field When a uniform magnetic field is applied, the particle magnetic moments of DHSs interact with each other and with the field. This leads to appearance of extra degrees of freedom due to the dipolar orientation coupling to the applied field. It significantly influences the structure of classical free energy virial expansion. A full derivation is given here, the structure of which follows closely Section 6 of Ref. [4]. Helmholtz free energy of DHS fluid under an applied magnetic field is
F = Fid + Fm − kT ln Q ,
Fm = −NkT ln
sinh α
α
,
(11)
where Fm is the free energy of ideal paramagnetic gas, Q stands for the configuration part of the partition function. Indicating the HS partition function as Qs , the third term on the right-hand side of (11) may be presented as follows
−kT ln Q = −kT ln Qs + △F ,
△F = −kT ln
Q Qs
.
(12)
Using the definition of the DHS partition function under an applied field, the △F can be written as
N ˆs ˆd H H 1 ˆ △F = −kT ln = −kT ln Ri exp − − Qs Qs i=1 kT kT
Q
N
ˆi
,
(13)
i=1
where ⟨· · ·⟩ˆ i is the averaging over all possible orientations of the ith magnetic moment with account for the dipole–field interaction
⟨1⟩ˆ i =
1
α
4π sinh α
ˆ d /kT In Eq. (13), exp −H
2π
0
π
1 · exp (α cos ωi ) sin ωi dωi dζi = 1.
(14)
0
is the expanded Taylor series
N
ˆ 1 ˆ i exp − Hs △F = −kT ln R Qs i=1
kT
1+
∞ p=1
1 p!
−
ˆd H kT
p N i=1
ˆi
.
(15)
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The first terms of the series (15) provide a major contribution to △F when the interparticle dipole–dipole interaction intensity is comparable with the thermal energy. In fact, in the general case, the number of terms under consideration depends on the type and intensity of interparticle interaction. Here the first, the second and the third term of the series are presented. ˆ i Rˆ j Ud (ij) = As all particles of the DHS model are identical, the contribution from particles i, j and 1, 2 into △F is similar: R
ˆ 1 Rˆ 2 Ud (12). Therefore, combinatorial counting of these terms is necessary [4]. R − 1
ˆd H kT
=
−
2!
N 1
kT i,j=1
ˆd H
−
2
2!
N (N − 1)
⇒
kT 1
=
kT
Ud (ij)
N
2
−
Ud (12) − ,
(16)
kT
2
Ud (ij)
=
1
2 N Ud (ij) −
kT 2! i
i ,j = 1 N
+
kT
i̸=j̸=m=1
N (N − 1)
⇒
+ 1
3!
−
ˆd H
3
kT
−
+
+
(kT )2
−
Ud (12)
3 +
kT
Ud (ij) kT
Ud (mp) − kT
N (N − 1)(N − 2) [−Ud (12)][−Ud (13)]
2!4
3!2
+
Ud (12)
2
−
i̸=j̸=m̸=p=1
2!2 kT 2! N (N − 1)(N − 2)(N − 3) [−Ud (12)][−Ud (34)]
N (N − 1)
⇒
kT
(kT )2 ,
(17)
3N (N − 1)(N − 2) [−Ud (12)]2 [−Ud (13)]
(kT )3
3!
N (N − 1)(N − 2) [−Ud (12)][−Ud (13)][−Ud (23)] 3! (kT )3 3N (N − 1)(N − 2)(N − 3) [−Ud (12)][−Ud (34)]2
(kT )3
3!4
+ ···.
(18)
ˆ d /kT )3 is presented up to the fourth order in terms of the particle number N 4 . Each factor of Above, the Hamiltonian (H −Ud (ij)/kT yields an additional power of the dipolar coupling constant λ. Using the expressions (16)–(18) for evaluation of △F (15) up to the order λ3 N 4 , and collecting together all terms of equal order in λ, we get △F = −kT ln 1 + A(λ) + B(λ2 ) + C (λ3 ) .
(19)
Here, the terms in the logarithm argument are clearly defined as functions of λ. However, these functions also depend on the density n, the number of particles N and the system volume V . The exact definitions of the terms are presented in Appendix A. Performing cumulant expansion (Section 6 of Ref. [4]) of the logarithm (19) within the accuracy of λ3 , we obtain
A2 (λ) 1 △F = −kT A(λ) + B(λ2 ) + C (λ3 ) − − A(λ)B(λ2 ) + A3 (λ) + · · · . 2
(20)
3
Since each term (20) is density dependent, we may regroup △F into series over n collecting terms with equal order of n
△F = −NkT
∞
(fds)
Bk+1 (λ)nk .
(21)
k=1
(fds)
(fds)
The regrouping procedure is derived in Appendix B for coefficients B2 (fds)
classical virial series (3), and the coefficients Bi (fds)
B3
and B3
in detail. Series (21) is analogous to the
(λ) are similar to the virial coefficients [4,5]. Coefficients B(2fds) (λ) and
(λ) up to terms of order of λ3 are given by (fds)
B2
(fds)
B3
= D1(f ) + D2(f ) + D3(f ) , (f )
= 3 · D4
(f )
+ 3 · D5
(22) (f )
+ 3 · D6
(f )
+ 3 · D7
+ 6 · D8
(f )
(f )
+ D9
− D10
(f )
− 2 · D11
(f )
(23)
where formulas for Di(f ) are shown in Tables 3 and 4. The expression and the diagram presentation for the second virial coefficient under an applied field is the same as for the zero-field case. The difference is only in the averaging operators ⟨· · ·⟩ˆ i
⟨· · ·⟩i . The structure of the third virial coefficient under an applied field B(3fds) differs from the case of zero field: unlike B(3ds) , (fds)
the coefficient B3 contains two additional diagrams D10(f ) and D11(f ) ; also the structure of diagrams D7(f ) and D8(f ) differs from that of D7 and D8. Nevertheless, (23) is the complete formula for the third virial coefficient within the accuracy of λ3 , and covers zero-, finite- and infinite-field cases. In zero-field (α = 0) or in an infinite field (α = ∞), 3 · D7(f ) − D10(f ) = D7 and 6 · D8(f ) − 2 · D11(f ) = D8. The proof is below.
E.A. Elfimova et al. / Physica A 415 (2014) 210–219
215
Table 3 (fds)
Diagrams corresponding to the second virial coefficient B2 DHS fluid under an applied field. Diagram (f )
D1
D2(f )
Coefficient
Formula
1
V 2
1
D3(f )
1
ˆ 12 − R
Ud (12) kT
up to the Ud3 -term for
ˆ 1 ˆ2
[fs (12) + 1]
V 2·2!
2 ˆ 12 − Ud (12) R kT
[fs (12) + 1]
V 2·3!
3 ˆ 12 − Ud (12) R kT
[fs (12) + 1]
ˆ2 ˆ 1
ˆ 1 ˆ2
Table 4 (fds)
up to the Ud3 -term for DHS fluid in an applied field.
Diagrams corresponding to the third virial coefficient B3 Diagram
Coefficient
Formula
D4(f )
3
V2 6
D5(f )
3
V2 R R 6·2! 12 13
D6(f )
3
ˆ 12 Rˆ 13 − R
ˆ
ˆ
V2 R R 6·3! 12 13
ˆ
ˆ
Ud (12) kT
ˆ 1 ˆ2
(12) − UdkT
2
(12) − UdkT
3
[fs (12) + 1]fs (13)fs (23)
[fs (12) + 1]fs (13)fs (23) ˆ 1 ˆ2
[fs (12) + 1]fs (13)fs (23) ˆ 1 ˆ2
[−Ud (12)][−Ud (13)] (kT )2 ˆ 1 ˆ 2 ˆ3
D7(f )
3
V2 6
D8(f )
6
V2 R R 6·2! 12 13
D9(f )
1
V2 6
ˆ 12 Rˆ 13 R
[−Ud (12)][−Ud (13)][−Ud (23)] (kT )3 ˆ 1 ˆ 2 ˆ3
D10(f )
1
V2 2
ˆ 12 Rˆ 34 R
[−Ud (12)][−Ud (34)] (kT )2 ˆ 1 ˆ 2 ˆ 3 ˆ4
D11(f )
2
V2 R R 2·2! 12 34
ˆ 12 Rˆ 13 R
ˆ
ˆ
ˆ
ˆ
[fs (12) + 1][fs (13) + 1][fs (23) + 1]
[−Ud (12)]2 [−Ud (13)] (kT )3 ˆ 1 ˆ 2 ˆ3
[fs (12) + 1][fs (13) + 1][fs (23) + 1]
[fs (12) + 1][fs (13) + 1][fs (23) + 1]
[fs (12) + 1][fs (34) + 1]
[−Ud (12)]2 [−Ud (34)] (kT )3 ˆ 1 ˆ 2 ˆ 3 ˆ4
[fs (12) + 1][fs (34) + 1]
Diagram D7(f ) may be presented as follows D7(f ) =
[−Ud (12)][−Ud (13)] [fs (12) + 1][fs (13) + 1]fs (23) 6 (kT )2 ˆ 1 ˆ 2 ˆ3 V2 [−Ud (12)][−Ud (13)] ˆ ˆ + R12 R13 [fs (12) + 1][fs (13) + 1]. 6 (kT )2 ˆ 1 ˆ 2 ˆ3 V2
ˆ 12 Rˆ 13 R
(24)
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E.A. Elfimova et al. / Physica A 415 (2014) 210–219
The first term (24) corresponds exactly to the diagram D7. The second term (24) will be completely compensated by the diagram D10(f ) in zero- and infinite-field, because in these cases
[−Ud (12)][−Ud (13)] (kT )2
Ud (12) Ud (34) ≡ − − .
kT
123
kT
12
34
Under the finite applied field, the averaging over magnetic moment orientation depends on the angle between the radius vector of the particle position and the field direction, therefore
[−Ud (12)][−Ud (13)] (kT )2
̸= −
Ud (12) kT
ˆ3 ˆ 2 ˆ 1
−
Ud (34) kT
ˆ2 ˆ 1
, ˆ4 ˆ 3
so the second term (24) and the diagram D10(f ) give non-zero contributions to the third virial coefficient. Similar discussion may be applied to the diagrams D8(f ) and D11(f ) . The main conclusion from this section is the following: the expression for the DHS second virial coefficient under an applied field is the same as in the zero- and infinite-field cases; in the finite fields, the structure of the DHS third virial coefficient differs from the case of zero- and infinite-fields. The exact expression for the third virial coefficient within the accuracy of λ3 , covering zero-, finite-, and infinite-field cases, is given by (23). 5. Initial magnetic susceptibility According to the known models [17–19], the initial magnetic susceptibility of the DHS fluid is well described by the following expression
4π χL (4π χL )2 χ = χL 1 + + , 3
(25)
144
where χL = 2ϕλ/π is the Langevin susceptibility. This expression, obtained with different methods, has been proved as giving excellent agreement with the computer simulations both for the ferrofluids [20] and the ferrogels [21]. (fds) (fds) To verify the correctness of the formulas for B2 and B3 , the initial magnetic susceptibility is calculated using the free energy virial expansion
1 m 2 ∂ 2 F , χ = lim − α→0 V kT ∂α 2
(26)
(fds)
F = Fid + Fm − kT ln Qs − NkT B2
(fds) 2
n + B3
n
.
The terms Fid and kT ln Qs do not depend on the Langevin parameter α , therefore they do not contribute to χ . The contribution from the free energy of the ideal paramagnetic gas Fm to χ is equal to χL . It is the first term in (25). (fds) (fds) ˆ j . The results of the To calculate B2 (22) and B3 (23) it is necessary to perform the integration over ⟨· · ·⟩ˆ i and R
ˆ j is described in orientational averagings ⟨· · ·⟩ˆ i are given in Ref. [22]. A procedure of averaging over particle positions R (fds)
Refs. [22,23] in detail. The explicit expressions for B2 (fds)
B2
= D1(f ) + D2(f ) = 4vλL2 (α) +
4v 3
(fds)
and B3
within the accuracy of λ2 are as follows
L2 (α) λ2 1 + 3 , 5
(fds)
= 3 · D4(f ) + 3 · D5(f ) + 3 · D7(f ) − D10(f ) , 1 L23 (α) (f ) (f ) 2 2 D4 = 0, D5 = v λ 2 ln 2 + 1+ 3 5 L(α) L(α) D7(f ) = 2v 2 λ2 L2 (α) 3 − 6 + 2v 2 λ2 L2 (α) 16 − 32 α α B3
D10(f ) = 32v 2 λ2 L4 (α) L(α) = coth α − 1/α,
L3 (α) = 1 − (fds)
3L(α)
α
(27) (28)
.
The contribution from −NkTB2 to χ is equal to 4π χL2 /3. It coincides with the second term in (25). The result for D7(f ) is divided into two terms, similar to (24), where the second term together with D10(f ) is an extra contribution to the third virial coefficient, conditioned by an applied magnetic field. It is easy to see that their sum gives a (fds) non-zero contribution to χ , and this is equal to χL (4π χL )2 /9. Finally, the total contribution from −NkTB3 to χ is equal to χL (4π χL )2 /144. It coincides with the third term in (25).
E.A. Elfimova et al. / Physica A 415 (2014) 210–219
217
6. Conclusion Expressions for the second and third coefficients of DHS free energy virial expansion are derived for the case of an applied magnetic field of various strength. Virial coefficients are presented in a form of series over the dipolar coupling constant λ. The formulas obtained have the accuracy of λ3 . Truncation at this level is accurate for physically realistic parameters λ ≃ 1. A diagram interpretation of the virial coefficients is offered. The DHS second virial coefficient has the same structure in zero-, finite-, and infinite-fields. The field-dependent DHS third virial coefficient differs from the cases of zero- and infinitefields, and the derivation is presented. Complete formula for the third virial coefficient within the accuracy of λ3 , covering zero-, finite-, and infinite-fields, is given. In comparison with normally-quoted result for the third virial coefficient of simple fluids the complete formula takes into account additional contribution, which compensates for each other in the zero- and infinite-field cases, but are not equal to zero in the finite-field case. The field-dependent second and third virial coefficients are calculated within the accuracy of λ2 . They are applied to the initial magnetic susceptibility of a DHS fluid. The correct result for the initial magnetic susceptibility, coinciding with those of Refs. [17–19], can be obtained only under the condition that the additional contribution in the third virial coefficient is taken into account. Acknowledgement The research is supported by the Ministry of Education and Science of the Russian Federation grant N 3.12.2014/K and contract N 02.A03.21.0006 from 27.03.2013. Appendix A
A(λ) =
=
N (N − 1) 2 N (N − 1) 2
ˆ 1 Rˆ 2 − R
ˆ 1 Rˆ 2 − R
Ud (12) kT Ud (12)
N 1
ˆ 1 ˆ2
kT
Qs i=3
ˆ i exp − R
ˆs H
kT
(s)
ˆ 1 ˆ2
g2 (r ),
(A.1)
B(λ2 ) = B1(λ2 ) + B2(λ2 ) + B3(λ2 ), B1(λ2 ) =
=
N (N − 1) 2!2 N (N − 1) 4
ˆ 1 Rˆ 2 R ˆ 1 Rˆ 2 R
− −
(A.2)
Ud (12)
2
kT
1
ˆ 1 ˆ2
Ud (12)
2
kT
N
Qs i=3
ˆ i exp − R
ˆs H
kT
(s)
g2 (r ),
ˆ 1 ˆ2
N ˆs H [− Ud (12)][−Ud (13)] 1 ˆ 1 Rˆ 2 Rˆ 3 ˆ i exp − R R B2(λ ) = 2! (kT )2 kT ˆ ˆ ˆ Qs i=4 1 2 3 N (N − 1)(N − 2) [− Ud (12)][−Ud (13)] (s) ˆ 1 Rˆ 2 Rˆ 3 R = g3 (r ), 2 (kT )2 ˆ 1 ˆ 2 ˆ3 N ˆs [− Ud (12)][−Ud (34)] 1 H N (N − 1)(N − 2)(N − 3) 2 ˆ 1 Rˆ 2 Rˆ 3 Rˆ 4 ˆ i exp − R R B3(λ ) = 2!4 (kT )2 kT ˆ 1 ˆ 2 ˆ 3 ˆ 4 Qs i=5 [− Ud (12)][−Ud (34)] N (N − 1)(N − 2)(N − 3) (s) ˆ 1 Rˆ 2 Rˆ 3 Rˆ 4 = R g4 (r ), 8 (kT )2 ˆ 1 ˆ 2 ˆ 3 ˆ4 N (N − 1)(N − 2)
2
C (λ3 ) = C 1(λ3 ) + C 2(λ3 ) + C 3(λ3 ) + C 4(λ3 ) C 1(λ3 ) =
=
C 2(λ3 ) =
N (N − 1) 3!2 N (N − 1) 12
ˆ 1 Rˆ 2 R ˆ 1 Rˆ 2 R
− −
3N (N − 1)(N − 2)
Ud (12)
3
kT
ˆ 1 ˆ2
Ud (12)
3
kT
ˆ 1 Rˆ 2 Rˆ 3 R
(A.3)
N ˆ 1 ˆRi exp − Hs
Qs i=3
kT
(s)
g2 (r ),
ˆ 1 ˆ2
[−Ud (12)]2 [−Ud (13)] (kT )3
3! ˆ ˆ ˆ 1 2 3 N 2 ˆ 1 (s) ˆ i exp − Hs N (N − 1)(N − 2) Rˆ 1 Rˆ 2 Rˆ 3 [−Ud (12)] [−Ud (13)] × R g3 (r ), Qs i=4 kT 2 (kT )3 ˆ ˆ ˆ 1 2 3
218
E.A. Elfimova et al. / Physica A 415 (2014) 210–219
N ˆs [−Ud (12)][−Ud (13)][−Ud (23)] 1 H ˆ ˆ ˆ ˆ Ri exp − C 3(λ ) = R1 R2 R3 3! (kT )3 kT ˆ ˆ ˆ Qs i=4 1 2 3 N (N − 1)(N − 2) (s) ˆ 1 Rˆ 2 Rˆ 3 [−Ud (12)][−Ud (13)][−Ud (23)] g3 (r ), R = 6 (kT )3 ˆ ˆ ˆ 1 2 3 N 2 ˆs 3N (N − 1)(N − 2)(N − 3) [−Ud (12)][−Ud (34)] 1 H 3 ˆ ˆ ˆ ˆ ˆ C 4(λ ) = Ri exp − R1 R2 R3 R4 3!4 (kT )3 kT ˆ 4 Qs i=5 ˆ 3 ˆ 2 ˆ 1 [− Ud (12)][−Ud (34)]2 N (N − 1)(N − 2)(N − 3) (s) ˆ 1 Rˆ 2 Rˆ 3 Rˆ 4 g4 (r ), R = 8 (kT )3 ˆ4 ˆ 3 ˆ 2 ˆ 1 N ˆs 1 H (s) ˆ gk = Ri exp − . N (N − 1)(N − 2)
3
Qs i=k+1
kT
(A.4)
Appendix B (s)
Notice, the terms A(λ), B(λ2 ) and C (λ3 ) contain a partial distribution function of HS fluid gi , which can be presented as a series in terms of density n [4] (s)
ˆ 3 fs (13)fs (23)) + O(n2 )], g2 (r ) = (fs (12) + 1)[1 + nV (R
(B.5)
(s)
g3 (r ) = (fs (12) + 1)(fs (13) + 1)(fs (23) + 1) + O(n)
(B.6)
(s)
g4 (r ) = (fs (12) + 1)(fs (13) + 1)(fs (14) + 1)(fs (23) + 1)(fs (24) + 1)(fs (34) + 1) + O(n).
(B.7)
ˆ j , (j ̸= 1) may be replaced by Rˆ 1j with particle 1 at the origin; in this case averaging Rˆ 1 yields 1. Averaging R • Coefficient B(2fds) (λ). (s)
In the lowest order of density in g2 (r ), the terms A(λ), B1(λ2 ) and C 1(λ3 ) have the orders of Nnλ, Nnλ2 and Nnλ3 , respectively. Here, the approximation of the thermodynamic limit (N − 1)/V = n is used. The remaining terms of (20) have (fds) in thermodynamic limit the orders higher than λ3 or higher than n. Thus, B2 (λ) has a form (fds)
B2
(λ) =
V 2
2 3 ˆR12 − Ud (12) + 1 − Ud (12) + 1 − Ud (12) kT
2
kT
6
kT
[fs (12) + 1],
(B.8)
ˆ 1 ˆ2
ˆ 1 Rˆ 2 is replaced by one Rˆ 12 with particle 1 at the origin. where integration over R • Coefficient B(3fds) (λ). (s)
In the first order of density in g2 (r ) the same terms A(λ), B1(λ2 ), and C 1(λ3 ) are proportional to n2 in thermodynamic (fds)
limit. Thus, they yield contribution to the coefficient B3
ˆ 12 Rˆ 13 − A(λ) + B1(λ ) + C 1(λ ) ⇒ Nn V R 2
3
2
2
(λ)
1 Ud (12) 2
kT
+
1 4
−
Ud (12) kT
2 +
1 12
−
Ud (12) kT
3 ˆ 1 ˆ2
× [fs (12) + 1]fs (13)fs (23). (s)
(B.9)
In the lowest order of density in g3 (r ), terms B2(λ ), C 2(λ ), and C 3(λ ) have the orders of Nn λ , Nn λ , and Nn2 λ3 , respectively, in approximation of the thermodynamic limit (N − 1)(N − 2)/V 2 = n2 . Thus, they also yield a contribution to (fds) the coefficient B3 (λ) 2
3
3
ˆ 12 Rˆ 13 [fs (12) + 1][fs (13) + 1][fs (23) + 1] B2(λ2 ) + C 2(λ3 ) + C 3(λ3 ) ⇒ Nn2 V 2 R
2 2
2 3
1 [−Ud (12)][−Ud (13)]
(kT )2 1 [−Ud2 (12)][−Ud (13)] 1 [−Ud (12)][−Ud (13)][−Ud (23)] . + + 2 (kT )3 6 (kT )3 ˆ 1 ˆ 2 ˆ3 2
(B.10)
Terms B3(λ2 ) and A2 (λ)/2 should be considered together, because in the thermodynamic limit and in the lowest order (s) (s) of density in g4 (r ) and g2 (r ) each term has an order of N 2 n2 λ2 , which indicates the divergency ∼ N 2 ; the sum of them,
E.A. Elfimova et al. / Physica A 415 (2014) 210–219
219
however, has the contribution of the order of Nn2 λ2 B3(λ ) − A (λ)/2 ⇒ 2
2
N (N − 1)(N − 2)(N − 3) − N 2 (N − 1)2
8
[−Ud (12)][−Ud (34)] [fs (12) + 1][fs (34) + 1] (kT )2 ˆ4 ˆ 3 ˆ 2 ˆ 1 V 2 Nn2 ˆ 12 Rˆ 34 [−Ud (12)][−Ud (34)] ⇒− [fs (12) + 1][fs (34) + 1], R 2 (kT )2 ˆ4 ˆ 3 ˆ 2 ˆ 1
× Rˆ 12 Rˆ 34
(B.11)
ˆ 12 and Rˆ 34 are performed independently in different frames with particles 1 and 3 at the origin, here, the averagings R respectively. Similarly, the terms C 4(λ3 ) and A(λ)B1(λ2 ) have divergent of ∼ N 2 separately, whereas their sum in the thermodynamic (s) (s) limit and in the lower order of partial distribution functions g4 (r ) and g2 (r ) defines the term of order Nn2 λ3 C 4(λ ) − A(λ)B1(λ ) ⇒ − 2
2
V 2 Nn2 2
2 ˆR12 Rˆ 34 [fs (12) + 1][fs (34) + 1] [−Ud (12)][−Ud (34)] . (kT )3 ˆ 1 ˆ 2 ˆ 3 ˆ4
(B.12)
In the thermodynamic limit approximation the remaining terms of (20) have the orders higher than n2 or higher than λ3 . (fds) Collecting terms (B.9)–(B.12), the coefficient B3 (λ) has a form within the accuracy of λ3 (fds)
B3
(λ) = V Rˆ 12 2
+ Rˆ 13
ˆ 13 − R
1 Ud (12) 2
kT
+
1
4
1 [−Ud (12)][−Ud (13)]
−
+
Ud (12) kT
2 +
1 12
−
Ud (12) kT
3
[fs (12) + 1]fs (13)fs (23) ˆ 1 ˆ2
1 [−Ud2 (12)][−Ud (13)]
(kT )2 2 (kT )3 1 [−Ud (12)][−Ud (13)][−Ud (23)] + [fs (12) + 1][fs (13) + 1][fs (23) + 1] 6 (kT )3 ˆ 1 ˆ 2 ˆ3 [−Ud2 (12)][−Ud (34)] 1 [−Ud (12)][−Ud (34)] ˆ + − R34 [fs (12) + 1][fs (34) + 1] . 2 (kT )2 (kT )3 ˆ 1 ˆ 2 ˆ 3 ˆ4 2
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
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