Free-energy functional of the Debye-Hückel model of two-component plasmas

T. Blenski1∗ and R. Piron2

1
Laboratoire “Interactions, Dynamiques et Lasers”,
UMR 9222, CEA-CNRS-Université Paris-Saclay,
Centre d’Études de Saclay, F-91191 Gif-sur-Yvette Cedex, France. and
2
CEA, DAM, DIF, F-91297 Arpajon, France.
(Dated: April 24, 2017)
We present a generalization of the Debye-Hückel free-energy-density functional of simple fluids to
arXiv:1704.06502v1 [physics.plasm-ph] 21 Apr 2017

the case of two-component systems with arbitrary interaction potentials. It allows one to obtain the
two-component Debye-Hückel integral equations through its minimization with respect to the pair
correlation functions, leads to the correct form of the internal energy density, and fulfills the virial
theorem. It is based on our previous idea, proposed for the one-component Debye-Hückel approach,
and which was published recently [1]. We use the Debye-Kirkwood charging method in the same
way as in [1], in order to build an expression of the free-energy density functional. Main properties
of the two-component Debye-Hückel free energy are presented and discussed, including the virial
theorem in the case of long-range interaction potentials.

PACS numbers: 05.20.Jj

I. INTRODUCTION extent, this explains why this model is so commonly used

in plasma physics as well as in the physics of electrolytes.
The Debye-Hückel (DH) model was introduced in the Our interest in the DH models stems from our research
theory of electrolytes [2]. It is the linearized version on density effects in the equation of state and in radiative
of the Non-Linear Debye-Hückel (NLDH) or Poisson- properties of dense plasmas. We aim to describe these
Boltzmann model. Both the DH and NLDH models can effects using approximate but fully variational models of
be in principle extended to arbitrary interaction poten- atoms in plasmas. The ion and free electron correlations
tials, with many applications in the physics of classical and their impact on the atomic structure and dynam-
fluids, charged or not. ics should thus be taken into account while preserving
the variational character of the approach. Up to now
An important feature of the one-component DH model the most used models are still essentially based on the
is the mean-field screening of the interaction potential, ion-in-cell picture [12–14]. Some progress towards a vari-
that results in the decay of the correlation function, even ational formulation of quantum atom-in-plasma models
when the interaction potential has a Coulomb tail. This was achieved [15–22]. However, it relies on a very sim-
property is preserved in the two-component DH theory ple hypothesis on ion-ion correlations. Attempts to in-
of plasma or charged mixtures. In general the DH model clude ion correlations into atom-in-plasma models were
is valid in the low-coupling limit, i.e. when the interac- also proposed (see, for instance, [23–26]), but do not stem
tion energy is small compared to the kinetic energy of from a fully variational derivation.
particles. The DH theory is a relatively simple mean- The DH model is a natural candidate tool in order
field approach to classical fluids in comparison to the to include plasma ions and electron effects for relatively
Hyper-Netted Chain (HNC) [3] or Percus-Yevick [4] mod- weakly coupled plasma. The results presented in this
els, which account for part of the correlations. However, paper extend the variational free-energy density formula
several more involved theoretical studies also proceed by presented in [1] to the two-component case. The present
introducing corrections, using the low-coupling DH limit
DH theory can also be useful to construct variational
as a starting point (see, for instance, [5, 6]). Recently it approaches to models of two-component fluids using ex-
was also shown that, in the DH model, the energy and
pressions of the free-energy density as a functional of the
virial ”routes” are thermodynamically consistent, for any pair correlation functions. In the atom-in-plasma models
potential [7]. For these reasons, the two-component DH
such a variational expression can be used in a more gen-
model can be of interest not only for the plasma physics eral theory that also includes the ion electronic structure.
community (see, for instance, [8, 9]) but also in the
physics of electrolytes (see, for example [10, 11]). In the A two-component free-energy density functional is
case of a long-range attractive potential, the linearization available in the HNC theory [27, 28]. However, as it was
performed in the DH model allows one to circumvent the the case in the well-known one-component DH theory,
“classical catastrophe” of collapsing particles. To some an expression of the free-energy density as a functional
of the pair correlation functions has not been yet pro-
posed. The purpose of this paper is to present a brief
derivation of such an expression in the DH approxima-
∗ Corresponding author tion, which can be seen as the DH equivalent to the HNC
Electronic address: thomas.blenski@cea.fr free-energy density functional of Lado [27] (see also [28]).
 2

II. DEBYE-HÜCKEL INTEGRAL EQUATIONS h12 (r) = − βu12 (r)

OF A TWO-COMPONENT FLUID β
Z
− d3 r′ {u12 (|r − r′ |) (̺1 h11 (r′ ) + ̺2 h22 (r′ ))
2
Let us consider a two-component homogeneous fluid
+h12 (|r − r′ |) (̺1 u11 (r′ ) + ̺2 u22 (r′ ))} (8)
at equilibrium, at a temperature T = 1/(kB β). In the
case of plasmas or charged P liquids the fluid neutrality We define
R theFourier transform Fk of a function F (r)
results in the charge balance j ̺j zj = 0 where ̺j is the as Fk = d3 r F (r)eik.r . The DH equations have a

average particle density and zj the charge of the j specie, simple form in the Fourier space:
respectively. The particles interact through potentials
uij (r). Using the so called “Percus trick” (see [29], also h11;k + ̺1 h11;k βu11;k + ̺2 h12;k βu12;k = −βu11;k (9)
[30]) we obtain the following identity : h22;k + ̺1 h12;k βu12;k + ̺2 h22;k βu22;k = −βu22;k (10)
h12;k + ̺1 h11;k βu12;k + ̺2 h12;k βu22;k = −βu12;k (11)
̺(1:j) {{ϕn (r′ ) = uin (r′ )}; r}
= gij (r) (1) The solutions of these equations are:
̺j
where the gij (r) are the pair distribution functions of the 1 1 β + β 2 ̺2 u22;k
heq,11;k = − + (12)
homogeneous fluid, for the species i and j, and ̺(1:j) (r) ̺1 ̺1 β Dk
is the j-specie 1-body reduced density matrix for a non- 1 1 β + β 2 ̺1 u11;k
heq,22;k =− + (13)
homogeneous fluid, with an external potential ϕn (r′ ) act- ̺2 ̺2 β Dk
ing on each specie n. 1 β 2 ̺1 ̺2 u12;k
The DH model is obtained from the static linear re- heq,12;k =− (14)
̺1 ̺2 β Dk
sponse of the density to the external potential:
where we have defined:
̺(1:j) {uin (r′′ ); r} ≈ ̺j
Dk = 1 + β(̺1 u11;k + ̺2 u22;k )+ β 2 ̺1 ̺2 (u11;k u22;k − u212;k )
( )
δ̺(1:j) {{ϕn (r′′ )}; r} 
XZ 
3 ′ ′
+ d r uin (r ) (15)
n
δϕn (r′ ) 
{ϕn (r ′ )=0}
(2)
III. EXPRESSION FOR A FREE-ENERGY
The functional derivatives of the density are (Yvon equa- FUNCTIONAL OF A TWO-COMPONENT FLUID
tions):

1 δ̺(1:j) (r) We are looking for a functional of trial correlations

= − ̺(2:j) (r, r′ ) + ̺(1:j) (r) ̺(1:j) (r′ ) functions hij (r) which, when minimized with respect to
β δϕj (r′ )
these functions, gives the DH equations and, at the DH
− ̺(1:j) (r′ ) δ3 (r − r′ ) (3) equilibrium, has the value of the free-energy excess due
1 δ̺ (1:j)
(r) to the interactions. As in [1, 27, 31], we use the charging
= − ̺(1:j,1:n) (r, r′ ) + ̺(1:j) (r) ̺(1:n) (r′ ) procedure due to Debye and Kirkwood [32]. The charging
β δϕn6=j (r′ )
parameter ξ allows one to switch on the interaction po-
(4)
tentials from zero to their actual values uξij (r) = ξ uij (r).
From the above equations and the definition (see, for in- For an exact interacting system one gets from the grand
stance, [30]) of the correlation functions hij (r) + 1 = canonical statistical sum (see, for instance, [30]) the fol-
gij (r) one obtains the equations of the Debye-Hückel lowing exact expression for the free-energy excess per unit
model : volume:
X Z Aξeq {{̺i }, β, {uij (r)}}}
hij (r) = −βuij (r) − β ̺n d3 r′ {uin (r′ )hin (|r − r′ |)}
n
V  
(5) Z ξ ′Z
dξ 1 X ′ ′

3 ξ ξ
For the purpose of future considerations it is useful to = ′
d r ̺ i ̺ j h eq,ij (r)u ij (r) (16)
write the DH equations in the following symmetrical 0 ξ 2
i,j

form:
Z We require, as in [1, 27, 31], that in our approximate
h11 (r) = −βu11 (r) − β d3 r′ {̺1 u11 (r′ )h11 (|r − r′ |) two-component DH theory, the searched free-energy den-
sity functional gives, at the equilibrium, the value of
−̺2 u12 (r′ )h12 (|r − r′ |)} (6) Eq. (16), with DH equilibrium functions heq,ij (r) (or
Z their equivalent forms in the Fourier space). Moreover,
h22 (r) = −βu22 (r) − β d3 r′ {̺2 u22 (r′ )h22 (|r − r′ |) as in [1], we postulate that the searched functional, which
depends on arbitrary trial functions hij;k , can be written
−̺1 u12 (r′ )h12 (|r − r′ |)} (7) as follows in the Fourier space:
 3

( " !
A {{̺i }, β, {uij (r)}, {hij (r)}} d3 k fk 2 h211;k h211;k
Z
= ̺1 + βu11;k h11;k + ̺1 βu11;k + ̺2 βu12;k h11;k h12;k
V (2π)3 2 2 2
!
h222;k h222;k
+̺22 + βu22;k h22;k + ̺2 βu22;k + ̺1 βu12;k h22;k h12;k
2 2
!#)
h212;k h212;k 1
+2̺1 ̺2 + βu12;k h12;k + (̺1 βu11;k + ̺2 βu22;k )
2 2 2
(17)

where the function fk depends on all ̺i , {uij;k } and on β, obtain:

but not on the {hij;k }. It is easy to check that the func-
tional derivatives of A {{̺i }, β, {uij (r)}, {hij (r)}} with
respect to the {hij;k }, at fixed ̺i and {uij;k }, lead to the Aξeq {{̺i }, β, {uij }}}
DH Eqs. (9), (10) and (11).
We postulate that such a function fk exists and that V  
d3 k f k ̺1 2 ξ
Z
the functional of Eq. (17), taken at equilibrium, has the = h βuξ + ̺1 ̺2 hξeq,12;k βuξ12;k
value given in Eq (16). A relatively simple way to ob- (2π) 3 2 2 eq,11;k 11;k
tain fk , proving also its existence, is to use the solution ̺2 2 ξ

of the DH equations in the Fourier space: hξeq,ij;k . Us- + heq,22;k βuξ22;k (22)
2
ing Eqs. (12),(13),(14) and (15), we can recast the free
energy-density excess of Eq. (16) as follows:

Aξeq {{̺i }, β, {uij }}} Substituting the values of the equilibrium correlation
functions given in Eqs. (9), (10), (11) and (15), we get:
V  
Z ξ ′Z
dξ d3 k  1 X ξ′ ξ′

= ′
̺ i ̺ i h eq,ij;k u ij;k (18)
0 ξ (2π)3  2 i,j ( " !
Aξeq {{̺i }, β, {uij }}} d3 k fk 1
 Z
= 1− ξ
V (2π)3 2
 
Z ξ ′Z Dk
dξ d3 k  1  
ξ′ ξ′

= −β ̺ 1 u 11;k + ̺ 2 u 22;k +
!#)
0 ξ
′ (2π)3  2β β  1
− ̺1 uξ11;k + ̺2 uξ22;k 1+ ξ (23)
 ′ ′
  ′ ′ ′
  2 Dk
2
β ̺1 uξ11;k + ̺2 uξ22;k + 2β̺1 ̺2 uξ11;k uξ22;k − uξ22;k 

Dξ
′

k
(19) Comparing Eqs. (23) and (21) we get the searched ex-
ξ pression for the function fk :
dξ ′ d3 k
 
1 ′ ∂
Z Z h  
ξ′ ξ′
= −ξ β ̺ u
1 11;k + ̺ u
2 22;k
0 ξ′ (2π)3 2β ∂ξ ′
 ′ iio
− ln Dkξ (20)
  h  i
ln Dkξ − β ̺1 uξ11;k + ̺2 uξ22;k
fk = h    i
Changing the order of the integrations we immediately β 1 − D1ξ − β2 ̺1 uξ11;k + ̺2 uξ22;k 1 + 1
Dkξ
get: k
(24)
Aξeq {{̺i }, β, {uij }}} The expression for the free-energy density functional
Eq. (17) with the calculated expression for the function
V  fk , Eq. (24), is the main result of the present paper.
d3 k 1 h  ξ h  ii
Z
ξ ξ
= ln D k − β ̺ 1 u 11;k + ̺ 2 u 22;k
(2π)3 2β It is interesting to check whether we can recover from
(21) the obtained free-energy density functional Eq. (17) the
expression for the free-energy density functional of the
On the other hand, using the DH equations (9),(10) and one-component DH system of [1]. Indeed, if we suppress
(11) we can also calculate the value of the excess free- the interaction between the two different species: u12;k =
energy density at the equilibrium from Eq. (17). We 0 (which also implies heq,12;k = 0), and use Eq. (17) with
 4

Eq. (24), we get: Eq. (24), the function fk depends on ξ only via the
potentials uξij (r). Moreover, the functional derivatives of
Aξ {{̺i }, β, {uii }, {hii }}}
Aξ {{̺i }, β, {uξij (r)}, {hij (r)}}/V with respect to hij (r)
V are equal to zero at the DH equilibrium. One can thus
 P    
ξ ξ
Z 3
d k  1 i ln 1 + β̺ i u ii;k − β̺ i u ii;k write formally:
= P 2
(2π)3  β 2 i ̺i heq,ii;k uii;k
Aξeq {{̺i }, β, {uij (r)}}
!# 
V
"
2
X h ii;k hii;k 
× ̺2i + hii;k βuii;k + ̺i βuii;k

2 2 A{{̺i }, β, {uξij (r)}, {hij (r)}} 
i
 =
V

(25)

eq,ξ
 
 d A{{̺ }, β, {uξ′ (r)}, {hij (r)}} 
Z ξ 
where

′ i ij
= dξ

 dξ ′ V

−βuii;k 0  
eq,ξ ′
heq,ii;k = (26)
1 + ̺i βuii;k Z ξ Z
= dξ ′ d3 r
The obtained functional Eq. (25), when minimized  0
with respect to the trial functions hii;k , gives the one-  ∂ A{{̺ }, β, {uξ′ (r)}, {hij (r)}} 
 ′
i ij ∂uξ11 (r)
component DH equation for the system i. If we sub-

 ∂u11 V  ′ ∂ξ ′

stitute into Eq. (25) the equilibrium values of Eq. (26), eq,ξ
then the functional takes the value of the sum of the free ξ ′  ′

energies of the two non-interacting systems 1 and 2. ∂ A{{̺i }, β, {uij (r)}, {hij (r)}}  ∂uξ12 (r)
+
∂u12 V  ′ ∂ξ ′

The postulated formula Eq. (16) is equivalent to the eq,ξ
following formulas for the functional derivatives of the 
ξ′

A{{̺ }, β, {u (r)}, {h (r)}} ξ′
excess free-energy density functional with respect to the ∂ i ij ij
 ∂u22 (r) 
+

interaction potentials: ∂u22 V  ′ ∂ξ ′ 

eq,ξ
̺2

δ A {{̺i }, β, {uij (r)}, {hij (r)}}  (31)
= 1 heq,11 (r)
δu11 (r) V 
eq 2
(27) where the |eq,ξ denote values taken for {hij (r) =
2 hξeq,ij (r)}. Comparing Eqs. (30) and (31) and taking into

δ A {{̺i }, β, {uij (r)}, {hij (r)}}  ̺2
δu22 (r) V  = 2 heq,22 (r) account that they are valid for a large class of indepen-
eq
dent potentials uξij (r), we get Eqs. (27), (28), (29). These
(28)
 equations can be also verified directly using the explicit
δ A {{̺i }, β, {uij (r)}, {hij (r)}}  form of Eqs. (17) and Eq. (24) but the necessary algebra
= ̺1 ̺2 heq,12 (r)
δu12 (r) V 
eq is tedious.
(29)

where the |eq denote values taken for {hij (r) = heq,ij (r)}.
Eqs. (27),(28),(29) are exact for an exact two-component IV. INTERNAL ENERGY AND VIRIAL
system (see, for instance, [33]). In our approximate DH THEOREM IN THE CASE OF
model they result from Eq. (16). In order to prove it, we TWO-COMPONENT DH THEORY
rewrite Eq. (16) in a slightly different way, obtaining:
The internal-energy density, as defined in thermody-
Aξeq {{̺i }, β, {uij (r)}}
namics is:
V  
Z ξ ′Z !
ξ
dξ 1 X Ueq βAξeq βAξ 
 
∂ ∂
′ ′

3 ξ ξ
= ′
d r ̺ ̺ h
i j eq,ij (r)u ij (r) = = (32)
0 ξ 2 V ∂β V ∂β V 
eq

i,j
 
Z ξ ξ′
1 X ∂u (r)
Z 
As stems from Eqs. (17) and (24), βAξ /V is a functional
′
ij
= dξ ′ d3 r ̺i ̺j hξeq,ij (r) (30)
0 2
i,j
∂ξ ′  of {hij (r)}, {̺i }, and of the functions {βξuij (r)}. As a
consequence, we can write:
Let us consider the functional
Aξ {{̺i }, β, {uξij (r)}, {hij (r)}}/V as given by Eq. (17), ∂

βAξ

∂

Aξ

=ξ (33)
where we put uξij (r) instead of uij (r). As stems from ∂β V ∂ξ V
 5

We then have, in virtue of Eq. (16): steps analogous to Eqs. (19), (20) and (20) and get:
 
ξ
Ueq {{̺i }, β, {uij (r)}} Z 3
d k  X ̺i ̺j ξ

Pv {{̺i }, β, {uij }} = h eq,ij;k βu ij;k 
 V  (2π)3  i,j 2
Z 1 X
4π ∞ dk
  3
k ∂
Z
= d3 r ̺i ̺j hξeq,ij (r)uξij (r) (34) + [ln (Dk )
2  3 0 (2π) 3 2β ∂k
i,j

which corresponds to the exact expression of the internal- −β (̺1 u11;k + ̺2 u22;k )] (39)
energy density, with hξeq,ij (r) taken in the DH approxi-
mation to the equilibrium correlation functions. The first term in Eq. (39) can be identified from Eq. (34)
The virial theorem for the exact two-component clas- as the internal-energy density Ueq {{̺i }, β, {uij (r)}}/V ,
sical fluid at equilibrium can be derived from the statisti- calculated in the Fourier space. The second term in
cal sum in the grand canonical ensemble formalism. We Eq. (39) can be integrated by parts (one may check that
consider this ensemble in a finite volume and the virial the integrand has correct behaviors at k = 0 and at
pressure is calculated from the exact thermodynamic for- k −→ ∞). We then get:
mula, following the technique used in [34], Sec. 30. The
obtained expression for the total thermodynamic pres- Pv {{̺i }, β, {uij }}
sure is P = P0 +Pv , where P0 = (̺1 + ̺2 ) /β corresponds Ueq {{̺i }, β, {uij (r)}}
to the ideal gas formula, while: =
V
1 d3 k
Z
Pv {{̺i }, β, {uij }}} − 3 {ln (Dk ) − β (̺1 u11,k + ̺2 u22,k )} (40)
  2β (2π)
1 ∂
Z X 
=− d3 r ̺i ̺j geq,ij (r) r (uij (r)) (35) Finally, using Eq. (21) with ξ = 1, we obtain:
6 
i,j
∂r 
Pv {{̺i }, β, {uij }}
is the Clausius virial pressure. We recall that geq,ij (r) = Ueq {{̺i }, β, {uij (r)}} Aeq {{̺i }, β, {uij (r)}}
= −
1 + heq,ij (r). Since we have: V V
(41)
   
d ∂uij;k
Z
3 In order to calculate the pressure from the thermody-
d r r uij (r) = lim −3uij;k − k (36)
dr k→0 ∂k namic definition, let us consider a volume in which there
is one particle of the specie “1”: V1 = 1/̺1 . Using the no-
we get in the case of interaction potentials having tion of this volume, we can calculate the thermodynamic
Coulomb tails: pressure of the fluid as:
 
Z X ̺ ̺ ∂u (r) 
i j ij
Pth {{̺i }, β, {uij }}
d3 r r   
2 ∂r d Aeq {{̺i }, β, {uij (r)}}
=− V1
 
i,j
 2 dV1 V
 
X 4π̺i ̺j zi zj 2π X d Aeq {{̺i }, β, {uij (r)}}
= − lim = − lim 2 ̺j z j  = 0 = −V1 −
k→0 2k 2 k→0 k dV1 V
i,j j
Aeq {{̺i }, β, {uij (r)}}
(37) − (42)
V
P
due to the overall neutrality j ̺j zj = 0. Thus in the However, in the volume V1 = 1/̺1 , there are also parti-
case of potentials having Coulomb tails (plasmas or mix- cles of the specie “2”. Then, changing ̺1 by an infinitesi-
tures of charged liquids), the virial pressure becomes: mal value d̺1 , we also change ̺2 . Since the change of the
density ̺1 should not violate the overall neutrality of the
Pv {{̺i }, β, {uij }}} plasma or charged liquids, we have to fulfill the relation
  z1 ̺1 + z2 ̺2 = 0, where the charges z1 , z2 are related to
1 ∂u (r)
Z
the asymptotic behavior of the interaction potentials at
X 
ij
=− d3 r ̺i ̺j heq,ij (r)r (38)
6  ∂r  large distances r: u11 (r) ∼ = z12 /r, u12 (r) ∼
= z1 z2 /r and
i,j
u22 (r) ∼
= z22 /r. For this reason, the changes in ̺1 and ̺2
are not independent and we have:
We substitute the solution to the DH Eqs. (9),(10) and
(11) into Eq. (38), written in the the Fourier space, and d̺2 ̺1
make use of a relation similar to Eq. (37). We repeat the = (43)
d̺1 ̺2
 6

Then, the first term of the RHS of Eq. (42) can be rewrit- Eqs. (42),(45) and (46) confirm the equivalence between
ten as: the thermodynamic definition of pressure and the virial
  pressure formula in the case of the two-component DH
d Aeq {{̺i }, β, {uij (r)}} theory based on the free-energy density functional pro-
−V1 −
dV1 V posed in the present paper.
 
d A eq {{̺ i β, {uij (r)}}
},
= ̺1 (44)
d̺1 V
(1) V. CONCLUSION
Denoting by Pth {{̺i }, β, {uij }} the first term of the last
line in Eq. (42) we thus have :
Our Eqs. (17) and (24) give an explicit expression of
the free-energy density functional in the Debye-Hückel
 
(1) ∂ Aeq {{̺i }, β, {uij (r)}}
Pth {{̺i }, β, {uij }} = ̺1 (DH) approximation for two-component fluids, extending
∂̺1 V
  our previous results of [1]. These expressions allows one
∂ Aeq {{̺i }, β, {uij (r)}}
+ ̺2 to obtain the DH equations from a minimization proce-
∂̺2 V dure with respect to the pair correlation functions. In the
(45) two-component DH case, as in the corresponding HNC
case of [27, 28], requiring that the exact charging rela-
However, as stems from Eq. (21), βAeq /V depends on tion is respected in the approximate model allows one to
the {̺i } only through the variables {β̺i }, i.e. we have define a free-energy density functional that yields the cor-
βAeq /V = βAeq {{β̺i }, {uij (r)}}/V . For that reason we rect expression for the internal-energy density and fulfills
can write: the virial theorem in the case of long-range potentials.
(1)
Pth {{̺i }, β, {uij }}
   
∂ βAeq ∂ βAeq
= ̺1 + ̺2 Acknowledgments
∂ (β̺1 ) V ∂ (β̺2 ) V
   
∂ (β̺1 ) ∂ βAeq ∂ (β̺2 ) ∂ βAeq
= + One of the authors (TB) acknowledges funding from
β ∂ (β̺1 ) V β ∂ (β̺2 ) V
  the EURATOM research and training program 2014–
∂ βAeq Ueq {{̺i }, β, {uij }} 2018 in the framework of the ToIFE project of the EURO-
= = (46)
∂β V V fusion Consortium under grant agreement No. 633053.

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