Sine-Gordon Theory for the Equation of State of Classical Hard-Core Coulomb systems. III Loopwise Expansion

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by  Jean-Michel Caillol

Sine-Gordon Theory for the Equation of State of Classical Hard-Core Coulomb
systems. III Loopwise Expansion
Jean-Michel Caillol ∗
LPT - CNRS (UMR 8627)
4 Bat. 210, Universit´e de Paris Sud
0 F-91405 Orsay Cedex, France
0 (February 2, 2008)
2
We present an exact field theoretical representation of an ionic solution made of charged hard
n
spheres. The action of the field theory is obtained by performing a Hubbard-Stratonovich trans-
a
J form of the configurational Boltzmann factor. It is shown that the Stillinger-Lovett sum rules are
satisfied if and only if all the field correlation functions are short range functions. The mean field,
2
1 Gaussian and two-loops approximations of the theory are derived and discussed. The mean field
approximation for thefree energy constitutes an exact lower boundfor theexact free energy,while

] themeanfield pressureis anexact upperbound. Theone-loop orderapproximation isshown tobe
h
identical with the random phase approximation of the theory of liquids. Finally, at the two-loop
c
orderand in thepecular caseof therestricted primitivemodel, onerecoversresultsobtained in the
e
m framework of themode expansion theory.
-
t KEY WORDS: Coulomb fluids; Screening; Sine-Gordon action; Loop expansion.
a
t
s
.
t
a
m
- I. INTRODUCTION
d
n
Various ionic systems including electrolyte solutions,molten salts,and colloidscan be studied with a goodapprox-
o
c imation in the framework of the so-called primitive model (PM) which consists in a mixture of M species of charged
[ hard spheres (HS) which differ by their respective charges and (or) diameters.1 Of special interest is the restricted

primitive model (RPM) where M =2, the hard spheres have all the same diameter, and the cations and anions bear
2
opposite charges q. In manyinstances, we shallalso considerthe specialprimitive model(SPM), where the number
v
±
5 M of species as well as the charges are arbitrary but all the ions have the same diameter σ.
6 Inthetwofirstpartsofthiswork,publishedsomeyearsagoandhereafterreferredtoasIandII,wehaveestablished
4 an exactfield theoreticalrepresentationof the RPM.2,3 The actionof this field theory,which is obtained by applying
5 theKac-Siegert-Stratonovich-Hubbard-Edwards(KSSHE)4–9 transformtothe Coulombpotential,lookslikethe sine-
0 Gordon action to which it reduces in the limit of point-like ions,10–12 hence the slightly abusive title of this series of
3
papers. Nowadayswe prefer the acronymKSSHE to christenthe action. The extendedsine-Gordonactionderivedin
0
paperIfortheRPMisobtainedhereforageneralPM.TheregularizationoftheCoulombpotentialwhichisrequired
/
at todefine properlythe KSSHEtransformisobtainedby asmearingofthe chargeoverthe HSvolume. Amoregeneral
m treatment where a part of the Coulomb interaction is incorporated in the reference system is discussed in the review
of Brydges and Martin.11
-
d Thedevelopmentsofrefs. IandIIarebasedonacumulantexpansionofthegrandpartitionfunctionreorganizedin
n ascending powersof either the fugacity or the inversetemperature. In this wayone canobtain the exact low fugacity
o
andhigh temperature expansionsof the pressure andthe freeenergyof the RPM.Of coursethe expressionsobtained
c
in that manner are already known from the theory of liquids and were derived years ago in the framework of Mayer
:
v graph expansions.13,14 In the present paper we proceed differently. After having obtained the KSSHE action for the
i
X generalPM (see sec. II), we reorganizethe cumulant expansion by grouping some classes of Feynman diagrams. The
resulting loopwise expansion is explicitly computed up to order two in the number of loops. In the case of the RPM,
r
a thetwo-looporderfreeenergyturnsouttocoincidewithanexpressionderivedmorethanthirtyyearsagobyChandler
and Andersen15 in the framework of the so-called mode expansion theory.16,17 Reorganizing the loop-expansion in
ascending powers of the inverse temperature gives back the high-temperature expansions of paper II and ref.14
∗
e-mail: [email protected]
1
Our paper is organized in the following way. In next section II we show how to construct a well-defined KSSHE
transform by regularizing the Coulomb potential at short distances by means of a smearing of the charges inside the
volumes of the HS. In section III we establish the general relations between the charge correlation functions and the
correlations of the KSSHE field. From the known asymptotic behavior of the former one can deduce that of the
latter. The conclusion, which is detailed at length in section IV, is that the n-body correlations of the KSSHE field
are short-ranged functions; stated otherwise, the KSSHE field is a non-critical field. The so-called Stillinger-Lovett
sum rules, both for the homogeneous18 and the inhomogeneous fluid19 emerge as a consequence of this behavior. In
section V the mean field (MF) level of the theory is studied in detail. The MF free energy β is shown to be a
MF
A
strictlyconvexfunctionaloftheM partialdensities andto constitute arigorouslowerboundofthe exactfreeenergy.
Theformerpropertyexcludesafluid-fluidtransitionattheMFlevelwhilethelatterservestodefineanoptimizedMF
free energy by maximizing β with respect to the variations of the smearing functions. An explicit expression of
MF
A
the optimized β is obtained in the case of an homogeneous fluid. From the MF solution for the inhomogeneous
MF
A
system we also deduce the expressions of the n-body correlation and vertex functions of the homogeneous system in
the Gaussianapproximation.20 This Gaussianapproximationis discussedinsectionVI andshownto be equivalentto
therandomphaseapproximation(RPA)ofthetheoryofliquids.1,21 Finally,atwo-loopordercalculationisperformed
insectionVII.TheresultingexpressionforthethefreeenergyoftheRPMisshowntobeidenticalwiththatobtained
by Chandler and Andersen15 in the framework of the first version of the mode expansion theory. Conclusions are
drawn in section VIII.
II. THE KSSHE TRANSFORM
A. The model
We shallconsideronly the three dimensional(3D) versionof the (PM),i.e. amixture of M species ofchargedhard
spheres.1 The ions of the species α (α = 1,...,M) are characterized by their diameter σ and their electric charge
α
q . Themolecularstructureofthesolventisignoredanditistreatedasacontinuum,thedielectricconstantofwhich
α
has been absorbed in the definition of the charges q . The solution is made of both positive and negative ions so
α
that the electroneutrality in the bulk can be satisfied without adding any unphysical neutralizing background to the
R
system. The particles occupy a domain Ω 3 of volume Ω of the ordinary space with free boundary conditions.
Only configurations ω (N ;~r1,...,~r1 ..⊂. N ;~rα,...,~rα N ;~rM,...,~rM ) (~rα Ω ) without overlaps of the
≡ 1 1 N1| | α 1 Nα| M 1 NM iα ∈
spheres-i.e. suchthat ~rα ~rβ (σ +σ )/2-docontributetothe partitionorgrandpartitionfunctions. Insuch
k iα− iβk≥ α β
a configuration, the charge q of each ion can be smeared out inside its volume according to a spherically symmetric
α
distributionq τ (r) withoutalteringthe configurationalenergyasaconsequenceofGausstheorem. The distribution
α α
τ (r) is a priori arbitrary,provided it satisfies the following properties :
α
τ (r)=0 if r σ σ /2, (2.1a)
α α α
≥ ≡
d3~r τ (r)=1. (2.1b)
α
Z
The electrostatic interaction energy of two charge distributions τ and τ the centers of which are located at the
α β
points ~r and~r of Ω respectively will be noted w (1,2). It reads as
1 2 α,β
wα,β(1,2)= d3r1′ d3r2′ τα( ~r1 ~r1′ )vc( ~r1′ ~r2′ )τβ( ~r2′ ~r2 ),
k − k k − k k − k
Z Z
τ (1,1′)v (1′,2′)τ (2′,2), (2.2)
α c β
≡
where v (r) = 1/r is the Coulomb potential. Note that in this paper, summation over repeated, either discrete or
c
continuous indices will always be meant (except if explicitly stated otherwise). As a consequence of eqs. (2.1) and of
Gausstheoremw (1,2)=1/r forr σ +σ . Note thatthe Fouriertransformw (k)ofthe interactiontakes
α,β 12 12 α β α,β
≥
the simple form
e
4π
w (k)= τ (k)τ (k), (2.3)
α,β k2 α β
which diverges for k 0 as 4π/k2 since τα(e0)=1, as folleows ferom eq. (2.1b). Finally we shall denote by
→
v (1,2)=q q w (1,2) (2.4)
e α,β α β α,β
2
the pair interaction of two ions.
The electrostaticpotentialenergyofthe configurationω timesthe inversetemperatureβ =1/kT canbe writtenas
β
βU (ω)= ρ (1)v (1,2)ρ (2) N νS , (2.5)
el 2 C c C − α α
where ρC(1) is the microscopic charge density inbthe configurabtion ω at the point ~r1 and ναS is the self-energy of the
charge distribution q τ (r). In general, for a sufficiently regular distribution τ (r), the self-energy
α α α
b
βq2 βq2 4π
νS = α w (0)= α d∨k τ (k)2 , (2.6)
α 2 α,α 2 k2 α
Z
whered∨k d3~k/(2π)3,isawell-definedpositiveandfinitequantity. OfecourseνS divergesforpoint-likechargeswhich
≡ α
makes the KSSHE transform, to be introduced in next section, an ill-defined object in that case. The microscopic
smeared charge density ρ (~r) which enters eq. (2.5) reads
C
ρ (1)=q τ (1,1′)ρ (1′), (2.7)
b C α α α
where
b b
Nα
ρ (1)= δ3(~r ~rα) (2.8)
α 1− iα
iXα=1
b
is the microscopic number density of the species α at the point~r .
1
It will prove convenient to make use of Dirac’s notations for matrix elements and scalar products and to rewrite
the energy (2.5) as
1 1
ρ (1)v (1,2)ρ (2)= ρ v ρ ,
C c C C c C
2 2 h | | i
= ρ V , (2.9)
b b Cb| b
D E
where V(1) ρ (1′) v (1′,1) denotes the microscopic electric pbotebntial at the point ~r in the configuration ω. Of
C c 1
≡
course V is solution of the 3D Poisson equation, i.e.
b b
b ∆1 V(1)= 4πρC(1). (2.10)
−
b b
B. The KSSHE transform of the Boltzmann factor
The Boltzmann factor in the configuration ω is equal to
exp( βU(ω))=exp( βU (ω)) exp( βU (ω)) , (2.11)
HS el
− − × −
where U (ω) denotes the contribution of the hard cores to the configurational energy. We perform now a KSSHE
HS
transform in order to rewrite eq. (2.11) as2–12
exp( βU(ω))=exp( βU (ω)) exp N νS exp iβ1/2 ρ ϕ , (2.12)
− − HS α α h C| i vc
(cid:0) (cid:1)D (cid:16) (cid:17)E
where the brackets ... denote Gaussian averagesover the real scalar field ϕ(~rb), i.e.
h ivc
1
... −1 ϕ...exp ϕv−1 ϕ ,
h ivc ≡Nvc D −2 | c |
Z (cid:18) (cid:19)
1 (cid:10) (cid:11)
ϕexp ϕv−1 ϕ , (2.13)
Nvc ≡ D −2 | c |
Z (cid:18) (cid:19)
(cid:10) (cid:11)
where
3
1
v−1(1,2)= ∆ δ(1,2) (2.14)
c −4π 1
is the inverse of the positive operator v (1,2). Therefore one has, after an integration by parts
c
1
= ϕ exp d3~r (~ϕ)2 (2.15)
Nvc D −8π ∇
Z (cid:18) ZΩ (cid:19)
The functional integrals which enter eqs (2.13) and (2.15) can be given a precise meaning when grounded perfect
conductorboundaryconditions(BC)areadopted;periodicBC’salsoworkifonlyneutralconfigurationsareconsidered,
we refer the reader to the literature for more details.2,11,20,22,23 It will be convenient to write
ρ ϕ = ρ φ , (2.16)
C α α
h | i h | i
where the smeared field φ is defined as
α b b
φ (1) β1/2 q τ (1,1′)ϕ(1′). (2.17)
α α α
≡
The field iφ (1) may thus be seen as an external one-body potential acting on the particles of the species α; indeed,
α
one can rewrite the Boltmann factor (2.12) under the form
exp( βU(ω)) =exp( βU (ω)) exp N νS exp(i ρ φ ) ,
− − HS α α h h α| αi ivc
(cid:0) (cid:1) M Nα
=exp(−βUHS(ω)) exp Nα ναS hexp b iφα ~riαα !ivc . (2.18)
(cid:0) (cid:1) αX=1iXα=1 (cid:0) (cid:1)
C. The Physical meaning of the auxiliary field
In a given configurationalω let us define an action
1
h[ϕ]= ϕv−1 ϕ i φ ρ , (2.19)
2h | c | i− h α| αi
and a partition function b
z(ω)= exp( h[ϕ]) . (2.20)
h − ivc
(Henceforth we shall specify the arguments of functionals by means of brackets, the variables of ordinary functions
being enclosed as usual by parenthesis.) The saddle point of the functional h[ϕ] is obtained by solving the equation
δh
=0, (2.21)
δϕ(~r)
(cid:12)ϕ
(cid:12)
(cid:12)
which can be recast under the form of the Poissonequa(cid:12)tion
∆ ϕ(1)= 4πiβ1/2ρ (1), (2.22)
1 C
−
the solution of which is of course
b
ϕ(1)=iβ1/2V(1). (2.23)
Therefore,atthesaddlepoint,thefieldϕcanbeidentifiedwiththemicroscopicelectricpotentialintheconfiguration
b
ω, up to an imaginary multiplicative constant. Moreover,it is easy to show that the value of h[ϕ] is nothing but the
energy of the configuration, i.e.
β
h[ϕ]= ρ v ρ . (2.24)
C c C
2 h | | i
Let us make now the change of variables ϕ = ϕ+δϕ wbhere δbϕ is a real scalar field. It follows from the stationarity
condition (2.21) that
4
1
h[ϕ]=h[ϕ]+ δϕv−1 δϕ , (2.25)
2 | c |
(cid:10) (cid:11)
which confirms that h[ϕ] is indeed a minimum of the functional h[ϕ] since v−1 is a positive operator. For a more
c
complicatedHamiltonianthanh[ϕ], theapproximationconsistingintruncatingitsfunctionalTaylorexpansionabout
the saddle point at the second order level is called the Gaussian approximation.20 This approximation is obviously
exact for z(ω) because h[ϕ] is a quadratic form. A direct calculation indeed confirms that
δϕexp 1 δϕv−1 δϕ
z(ω)=exp( h[ϕ]) D −2 | c | ,
− ϕexp 1 ϕv−1 ϕ
R D (cid:0)−2(cid:10) | c | (cid:11)(cid:1)
β
=exp ρ vR ρ .(cid:0) (cid:10) (cid:11)(cid:1) (2.26)
C c C
−2h | | i
(cid:18) (cid:19)
b b
D. The KSSHE transform of the grand partition function
Henceforwardwe shallwork in the grandcanonical(GC) ensemble. We denote by µ the chemicalpotential of the
α
species α andby ψ (~r)the externalpotentialwith whichthe particlesof the species α interacteventually. According
α
to a terminology due to J. Percus,24 we shall define the local chemical potential ν (~r) as β(µ ψ (~r)). With these
α α α
−
notations, the GC partition function of the system takes the form
∞ 1 ∞ 1 M Nα
Ξ[ ν ]= ... d3~r1...d3~rM exp( βU(ω)) exp ν ~rα . (2.27)
{ α} N ! N ! 1 NM − α iα
NX1=0 1 NXM=0 M ZΩ αY=1iYα=1 (cid:0) (cid:0) (cid:1)(cid:1)
Grand canonical averages of dynamic variables (ω) will be noted (ω) . Inserting the expression (2.18) of the
A hA iGC
Boltmann factor in eq. (2.27) one obtains the KSSHE representation of Ξ
Ξ[ ν ]= Ξ [ ν +iφ ] , (2.28)
{ α} h HS { α α}ivc
where ν = ν +νS and Ξ [ ν +iφ ] is the GC partition function of a mixture of bare hard spheres in the
α α α HS { α α}
presence of the local chemical potentials ν +iφ . The above result generalizes to the case of the PM the result
α α
obtained in paper I for the restricted primitive model. It is also possible to incorporate a part of the Coulomb
interaction in the reference potential which yields a more general expression than eq. (2.28) as detailed in the review
of Brydges and Martin (cf eq. (2.29) of ref11). However, in the liquid domain, the thermodynamics and correlations
of this reference system are, by contrast with those of the HS fluid, little known in general. Relations similar to eq.
(2.28) have also been obtained and discussed for neutral fluids.5,8,25,26
To make some contact with statistical field theory we introduce the effective Hamiltonian (or action)
1
[ϕ]= ϕv−1 ϕ logΞ [ ν +iφ ] , (2.29)
H 2 | c | − HS { α α}
(cid:10) (cid:11)
which allows us to recast Ξ under the form
Ξ[ ν ]= −1 ϕ exp( [ϕ]). (2.30)
{ α} Nvc D −H
Z
It will be important in the sequel to distinguish carefully, besides the GC averages < ...> , between two types
GC
of statistical field averages: the already defined <...> and the <...> that we define as
vc H
ϕ exp( [ϕ])A[ϕ]
<A[ϕ]> D −H . (2.31)
H
≡ ϕ exp( [ϕ])
R D −H
With these definitions in mind one notes that for anRarbitraryfunctional [ϕ] one has the relation
A
A[ϕ]Ξ [ ν +iφ ]
<A[ϕ]> = h HS { α α}ivc . (2.32)
H
Ξ [ ν +iφ ]
h HS { α α}ivc
5
III. CORRELATION FUNCTIONS
A. Zoology
The ordinary and truncated (or connected) density correlationfunctions will be defined in this paper as27,28
n
G(n) [ ν ](1,...,n)= ρ (i) ,
α1...αn { α} * αi +
1Y=1 GC
=Ξ[ ν b]−1 δn Ξ[{να}] ,
α
{ } δν (1)...δν (n)
α1 αn
δnlogΞ[ ν ]
G(n)T [ ν ](1,...,n)= { α} . (3.1)
α1...αn { α} δν (1)...δν (n)
α1 αn
Our notation emphasizes the fact that the G(n) (truncated or not) are functionals of the local chemical potentials
α1...αn
ν (~r) and functions of the coordinates (1,...,n) (~r ,...,~r ). Note however that, in the remainder of the paper,
α 1 n
≡
we shall frequently omit to quote the functional dependence of G(n) upon the ν when no ambiguity is possible.
α1...αn α
In standard textbooks of liquid theory1 the n-body correlation functions are more frequently defined as functional
derivatives of Ξ or logΞ with respect to the activities z = exp(ν ) rather than with respect to the local chemical
α α
potentials. This yields differences involving delta functions. For instance for n = 2 and for a homogeneous system
one has
G(2)(1,2)=ρ ρ g (r )+ρ δ δ(1,2),
αβ α β αβ 12 α α,β
G(2)T(1,2)=ρ ρ h (r )+ρ δ δ(1,2), (3.2)
αβ α β αβ 12 α α,β
where ρ is the equilibrium number density of the species α and g (r) the usual pair distribution function; finally
α αβ
h =g 1.
αβ αβ
−
The charge correlations will play an important role in subsequent developments. They are defined as
n
G(n)(1,...,n)= ρ (i) . (3.3)
C C
* +
1Y=1 GC
b
It follows from the definition (2.7) of the smeared density of charge ρ that eq. (3.3) can be rewritten alternatively
C
G(n)(1,...,n)=q ...q τ (1,1′)...τ (n,n′)G(n) (1′,...,n′). (3.4)
C α1 αn α1 αn b α1...αn
Clearly the operator
δ
Θ(1) iβ1/2q τ (1,1′) (3.5)
≡ α α δν (1′)
α
is the generator of the charge correlations for we have clearly
inβn/2 G(n)(1,...,n)=Ξ−1Θ(1)...Θ(n)Ξ. (3.6)
C
The truncated charge correlations can thus be defined according to
inβn/2 G(n)T(1,...,n)=Θ(1)...Θ(n)logΞ. (3.7)
C
On the one hand
G(n)T(1,...,n)=q ...q τ (1,1′)...τ (n,n′)G(n)T (1′,...,n′), (3.8)
C α1 αn α1 αn α1...αn
and, in the other hand22,27,28
G(n)T(1,...,n)=G(n)T(1,...,n) G(m)T(i ,...,i ), (3.9)
C C − C 1 m
m<n
X Y
6
where the sum of products is carried out over all possible partitions of the set (1,...,n) into subsets of cardinality
m<n. ThefunctionsG(n) (resp. G(n)T)fordifferentvaluesofnarenotindependent;theyarerelatedbyahierarchy
C C
of equations most conveniently written with the help of the operator Θ defined at eq. (3.5). The hierarchies for the
G(n) and the G(n)T are derived in appendix A.
C C
Inthe fieldtheoreticalrepresentationofthe PMthe fieldcorrelationfunctions playakeyrole. Theyaredefinedas
G(n)(1,...,n)= ϕ(1)...ϕ(n) , (3.10a)
ϕ h iH
G(n)T(1,...,n)=G(n)T(1,...,n) G(m)T(i ,...,i ). (3.10b)
ϕ ϕ − ϕ 1 m
m<n
X Y
OfcoursetheG(n),asthechargecorrelationfunctions,arefunctionalsofthelocalchemicalpotentials. Thehierarchies
ϕ
for the G(n) and the G(n)T are derived in appendix A.
ϕ ϕ
B. Relations between the charge and field correlation functions
1. The density and charge correlation functions as statistical field averages
It follows from the definition (3.1) of G(n) and from the KSSHE representation (2.28) of the grand partition
α1...αn
function that we have
1 δn Ξ [ ν +iφ ]
G(n) [ ν ](1,...,n)=Ξ−1 −1 ϕ exp ϕv−1 ϕ HS { α α} ,
α1...αn { α} Nvc D −2 | c | δν (1)...δν (n)
Z (cid:18) (cid:19) α1 αn
=Ξ−1 Ξ G(n) [ ν(cid:10) +iφ (cid:11)](1,...,n) , (3.11)
HS HS,α1...αn { α α} vc
D E
where G(n) [ ν +iφ ](1,...,n) denotes the density correlation function of the reference HS fluid in the
HS,α1...αn { α α}
presence of the local chemical potentials ν +iφ . Thence, making use of eq. (2.32)
α α
{ }
G(n) [ ν ](1,...,n)= G(n) [ ν +iφ ](1,...,n) . (3.12)
α1...αn { α} HS,α1...αn { α α} H
D E
Eq. (3.12), which extends to ionic mixtures a relation that we derived elsewhere for simple non-charged fluids,26
(n)
althoughaesthetic is notveryuseful sincethe hardspherecorrelationsG arecomplicatedfunctionals ofthe
HS,α1...αn
field ϕ. However the case n=1 is of some interest. In that case eq. (3.12) says that
ρ [ ν ](1)= ρ [ ν +iφ ](1) , (3.13)
β { α} h HS,β { α α} iH
It follows readily from the expression (3.4) of the charge correlation function that we also have
G(n)[ ν ](1,...,n)= G(n) [ ν +iφ ](1,...,n) , (3.14)
C { α} HS,C { α α} H
D E
where
G(n) (1,...,n)=q ...q τ (1,1′)...τ (n,n′)G(n) (1′,...,n′). (3.15)
HS,C α1 αn α1 αn HS,α1...αn
Specializing eq. (3.14) for n=1 we note that
ρ [ ν ](1)= ρ [ ν +iφ ](1) , (3.16)
C { α} h HS,C { α α} iH
where
ρ (1)=q τ (1,1′)ρ (1′). (3.17)
HS,C α α HS,α
7
2. Relations between G(n) and G(n)
C ϕ
Itfollowsreadilyfromtheexpression(3.7)ofG(n) andfromtheKSSHErepresentation(2.28)ofthegrandpartition
C
function that
1
inβn/2G(n)[ ν ](1,...,n)=Ξ−1[ ν ] −1 ϕ exp ϕv−1 ϕ
C { α} { α} Nvc D −2 | c | ×
Z (cid:18) (cid:19)
Θ(1)...Θ(n)Ξ [ ν +iφ ] .(cid:10) (cid:11) (3.18)
HS α α
× { }
At this point we make the remark that
δ Ξ [ ν +iφ ] δ iφ (1′) δ Ξ [ ν +iφ ]
HS α α α HS α α
{ } = { } ,
δ ϕ(1) δϕ(1) δ ν (1′)
α
=Θ(1)Ξ [ ν +iφ ] . (3.19)
HS α α
{ }
The relation (3.19) enables us to replace the operators Θ(i) which occur the right hand side (RHS) of eq. (3.18) by
functional derivatives with respect to the field ϕ. Then, performing n functional integrations by parts23,26 yields
δnexp 1 ϕv−1 ϕ
inβn/2G(n)(1,...,n)=( )n −2 | c | (3.20)
C − * δϕ((cid:0)1)..(cid:10).δϕ(n) (cid:11)(cid:1)+
H
The relation (3.20) can be used to obtain an explicit representation of G(n) in terms of the field correlations as long
C
as n is not too large. Let us first consider the case n=1 in which Eq. (3.20) takes the simple form
∆ <ϕ(1)> = 4πiβ1/2ρ (1). (3.21)
1 H C
−
Once again (cf eq. (2.22)) we obtain the Poissonequation, the solution of which is of course
<ϕ(1)> =iβ1/2V(1), (3.22)
H
where V(1) is the GC average of the configurational electric potential, i.e. V(1)=<V(1)> .
GC
In the case n=2 eq. (3.20) says that
b
1 1
βG(2)(1,2)= − ∆ δ(1,2) ∆ ∆ G(2)(1,2), (3.23)
C 4π 1 − (4π)2 1 2 ϕ
or, by reverting the equation
G(2)(1,2)=v (1,2) βv (1,1′)G(2)(1′,2′)v (2′,2). (3.24)
ϕ c − c C c
Eqs.(3.23)and(3.24)extendtoelectrolytesolutionsrelationsobtainedrecentlyforneutralfluids.26 Equationsofthis
type were also derived by Ciach and Stell in the framework of a heuristic field theory of the RPM.29
By combining eqs. (3.21) and (3.23) one can show easily that the truncated two-body charge correlation function
satisfies to a similar relation, i.e.
1 1
βG(2)T(1,2)= − ∆ δ(1,2) ∆ ∆ G(2)T(1,2). (3.25)
C 4π 1 − (4π)2 1 2 ϕ
In the case n=3 eq. (3.20) yields an awkwardexpressionfor G(3). However the truncated 3-body charge correlation
C
function takes the simple form
1
iβ3/2G(3)T(1,2,3)= ∆ ∆ ∆ G(3)T(1,2,3), (3.26)
C (4π)3 1 2 3 ϕ
which can be obtained by brute force calculation. The above result suggests that there are simple relations between
G(n)T and G(n)T for values of n 3. Indeed, let us apply the operator Θ(2)...Θ(3) (n 3) to both sides of eq.
C ϕ ≥ ≥
(n)T (n)T
(3.21). Then, making use of the hierarchy equations satisfied by the G and the G (cf eqs. (A1) and (A9) of
C ϕ
appendix A) one gets immediately the aesthetic generic formula
( 1)n
inβn/2G(n)T(1,...,n)= − ∆ ...∆ G(n)T(1,,...,n) ( n 3). (3.27)
C (4π)n 1 n ϕ ∀ ≥
8
3. Correlations of the electric potential
It is obvious that
V(1)...V(n) =v (1,1′)...v (n,n′)G(n)(1′,...,n′),
c c C
GC
D ET
Vb(1)...Vb(n) =v (1,1′)...v (n,n′)G(n)T(1′,...,n′). (3.28)
c c C
GC
D E
Combining the above relatbions witbh those obtained in sec. (IIIB2) one gets
iβ1/2 V(1) = ϕ(1) , (3.29a)
GC h iH
D ET
β V(1)Vb(2) =v (1,2) G(2)T(1,2), (3.29b)
GC c − ϕ
D ET
inβn/2 V(1)b...Vb(n) =G(n)T(1,,...,n) ( n 3). (3.29c)
GC ϕ ∀ ≥
D E
What can be learned from the abobve relatibons is the subject of next section.
IV. STILLINGER-LOVETT SUM RULES
A salient property of 3D ionic liquids is the screening effect. To paraphrase Ph. Martin, this type of fluid ” in
thermal equilibrium does not tolerate any charge inhomogeneity over more than a few intermolecular distances”.30
Evenattheliquid-vaporcriticalpointwherethecorrelationlengthassociatedwiththefluctuationsofdensitydiverges
it is believed, andhasbeen checkedby means ofnumericalsimulationsin the case ofthe RPM,31 that the correlation
length associated with the fluctuations of charge remains finite. From the existence of screening it is possible to
deduce sum rules for the charge correlation functions for both homogeneous and inhomogeneous systems, the so-
calledStillinger-Lovett(SL) sumrules.18,19,32 As pointedoutby B.Jancovici,these rulesmay be rederivedunder the
sole assumption that the system behaves macroscopically as a conductor in the sense that ” the laws of macroscopic
electrostatics are assumed to be obeyed for length scales large compared to the microscopic characteristic lengths of
the model.”33 We examine below how the SL rules can be deduced from simple hypothesis on the behavior of the
KSSHE field correlation functions.
Let us first consider a homogeneous system. In this case < ϕ > is a constant as well as are the densities ρ
H α
and the smeared density of charge ρ . It follows then from eq. (3.21) that the smeared density ρ = 0. This is
C C
nothing but the usual electroneutrality condition since ρ =ρ q τ (0) can be identified with the usual local charge
C α α α
density for τ (0) = 1 (property (2.1b) of the smearing function τ ). Therefore, in the framework of our formalism,
α α
the electroneutrality condition
e
e
ρ q =0 (4.1)
α α
isautomaticallysatisfiedforanarbitrarysetofchemicalpotentials ν ,awellknownpropertyofionicfluids.11 Note
α
{ }
that eq. (4.1) implies that there are only M 1 independent chemical potentials.
−
The correlations of the electric potential has been studied by various approaches and asymptotically one has
T
β V(1)V(2) =v (1,2), (4.2a)
c
GC
D ET
V(1b)...Vb(n) =0. (4.2b)
GC
D E
It must be stressed that these expressionbs are vablid for relative distances large compared to the microscopic charac-
teristic lengths of the system and if the correlations decay fast enough or, equivalently, if the system behaves as a
macroscopicconductor.11,30,33,34Thecomparisonoftheseasymptoticbehaviorswiththeexactrelations(3.29)derived
(n)T
in sec. (IIIB3) entails that the truncated KSSHE field correlation functions G are short range functions; stated
ϕ
otherwise, the KSSHE field is a non-critical field. Conversely, this property being taken as given, we show now that
one can infer the SL rules.
We consider now a non-homogeneous system and we take the Laplacian of eq. (3.29b). We get
9
T 1
β ρ (1)V(2) =δ(1,2)+ ∆ G(2)T(1,2). (4.3)
C GC 4π 1 ϕ
D E
Letusintegrateeq.(4.3)over~r . WbiththbehypothesisthatG(2)T(1,2)isshortrangetheintegrationoftheLaplacian
1 ϕ
gives zero by an application of Green’s theorem. Therefore
T
β d(1) ρ (1)V(2) =1 (4.4)
C
GC
Z D E
which is the Carnie-Chan sum rule.19 In the case ofba hombogeneous system, the Carnie-Chan sum rule is equivalent
to the SL sum rule.33 Let us retrieve these sum rules in our framework. We start with eq. (3.25) for a homogeneous
system. In this case G(2)T(1,2) G(2)T(~r =~r ~r )) and we have
C ≡ C 1− 2
1
4πβG(2)T(~r)=∆δ(~r) ∆∆G(2)T(~r). (4.5)
− C − 4π ϕ
With the hypothesis that G(2)T(1,2) is a short range function the integration over~r gives zero, i.e.
ϕ
d3~r G(2)T(~r)=0 (4.6)
C
Z
which is the first SL rule.18 Similarly, after integration by parts
6
4πβ d3~rr2G(2)T(~r)= 6+ d3~r ∆G(2)T(~r). (4.7)
C − 4π ϕ
Z Z
If G(2)T(~r) is a short range function then the integral in the RHS vanishes and we are left with the second SL sum
ϕ
rule
2πβ
d3~rr2G(2)T(~r)= 1. (4.8)
3 C −
Z
The SL rules are more conveniently written in terms of the pair distributions h introduced at eq. (3.2)
α,β
ρ q h (k =0)= q (4.9a)
γ γ α,γ α
−
2πβ
q q ρ ρ d3~r r2h (r)= 1 (4.9b)
α γ α γ e α,γ
3 −
Z
Althougheqs.(4.9b)and(4.8)areequivalent,eq.(4.9a)whichsaysthatthe cloudofchargeswhichsurroundsagiven
ion of species α has a total charge q implies eq. (4.6) but is more precise. It can nevertheless be derived directly
α
−
with methods similar to those used in this section. The proof is given in appendix B.
We want to precise that the results derived in this section are not valid for 2D systems which can undergo a
Kosterlitz-Thouless transition.35 In the low-temperature KT phase of a 2D PM, the SL rules are violated and the
sine-Gordon field should exhibit long range correlations with an algebraic decay. In this case, ϕ is a critical field and
<ϕ> is related to the order parameter of the KT transition.36
V. THE MEAN FIELD THEORY
A. Mean field equations
We define the MF level or saddle point approximationof the theory by the equation
Ξ [ ν ] exp( (ϕ)) , (5.1)
MF α
{ } ≡ −H
where,atϕ=ϕ,theaction isstationary. Itfollowsfromtheexpression (2.29)of thatthe stationaritycondition
H H
δ
H =0 (5.2)
δϕ(~r)
(cid:12)ϕ
(cid:12)
(cid:12)
(cid:12)
10