![Fig. 9. Fluid densities on isobar P = 0.85MPa as a function of temperature. The curve is the experimental data from [41] and the symbols are from path integral simulations [42] and from this work.](/figures/fig-9-fluid-densities-on-isobar-p-0-85mpa-as-a-function-of-319d74pv.webp)
![Table 2. Comparison of the pressure P resulting from our AstroPE FORTRAN-90 code with the pressure corresponding to Ree [17] MC simulations. The corresponding values of the reduced density n are listed. Quantum effects are not included.](/figures/table-2-comparison-of-the-pressure-p-resulting-from-our-2cers0qn.webp)
Article
Reference
Equation of state and stability of the helium-hydrogen mixture at
cryogenic temperature
SAFA, Yasser Fouad, PFENNIGER, Daniel
Abstract
The equation of state and the stability of the helium-molecular hydrogen mixture at cryogenic
temperature up to moderate pressure are studied by means of current molecular physics
methods and statistical mechanics perturbation theory. The phase separation, segregation
and hetero-coordination are investigated by calculating the Gibbs energy depending on the
mixture composition, pressure and temperature. Low temperature quantum effects are
incorporated via cumulant approximations of the Wigner-Kirkwood expansion. The interaction
between He and H2 is determined by Double Yukawa potentials. The equation of state is
derived from the hard sphere system by using the scaled particle theory. The behavior of the
mixture over a wide range of pressure is explored with the excess Gibbs energy of mixing and
the concentration fluctuations in the long wavelength limit. The theory is compared to
cryogenic data and Monte-Carlo calculation predictions. Contrary to previous similar works,
the present theory retrieves the main features of the mixture below 50 K, such as the critical
point and the condensation-freezing curve, and is found to [...]
SAFA, Yasser Fouad, PFENNIGER, Daniel. Equation of state and stability of the
helium-hydrogen mixture at cryogenic temperature. The European Physical Journal. B,
Condensed Matter, 2008, vol. 66, no. 3, p. 337-352
DOI : 10.1140/epjb/e2008-00438-8
Available at:
http://archive-ouverte.unige.ch/unige:112997
Disclaimer: layout of this document may differ from the published version.
1 / 1
Eur. Phys. J. B 66, 337–352 (2008)
DOI: 10.1140/epjb/e2008-00438-8
THE EUROPEAN
PHY SICAL JOURNAL B
Equation of state and stability of the helium-hydrogen mixture
at cryogenic temperature
Y. Safa
a
and D. Pfenniger
Geneva Observatory, University of Geneva, 1290 Sauverny, Switzerland
Received 11 April 2008 / Received in final form 15 October 2008
Published online 4 December 2008 –
c
EDP Sciences, Societ`a Italiana di Fisica, Springer-Verlag 2008
Abstract. The equation of state and the stability of the helium-molecular hydrogen mixture at cryogenic
temperature up to moderate pressure are studied by means of current molecular physics methods and
statistical mechanics perturbation theory. The phase separation, segregation and hetero-coordination are
investigated by calculating the Gibbs energy depending on the mixture composition, pressure and tem-
perature. Low temperature quantum effects are incorporated via cumulant approximations of the Wigner-
Kirkwood expansion. The interaction between He and H
2
is determined by Double Yukawa potentials. The
equation of state is derived from the hard sphere system by using the scaled particle theory. The behavior
of the mixture over a wide range of pressure is explored with the excess Gibbs energy of mixing and the
concentration fluctuations in the long wavelength limit. The theory is compared to cryogenic data and
Monte-Carlo calculation predictions. Contrary to previous similar works, the present theory retrieves the
main features of the mixture below 50 K, such as the critical point and the condensation-freezing curve,
and is found to be usable well below 50 K. However, the method does not distinguish the liquid from the
solid phase. The binary mixture is found to be unstable against species separation at low temperature
and low pressure corresponding to very cold interstellar medium conditions, essentially because H
2
alone
condenses at very low pressure and temperature, contrary to helium.
PACS. 64.10.+h General theory of equations of state and phase equilibria – 64.75.Gh Phase separation
and segregation in model systems (hard spheres, Lennard-Jones, etc.) – 64.75.Ef Mixing
1 Introduction
Studying the mixing behavior of helium (
4
He) and molec-
ular hydrogen (
1
H
2
) forming a binary system over a wide
range of conditions is extremely important for astrophysi-
cal purposes in view of the ubiquity of this mixture in the
Universe. As by-product, such as study is also of interest
for industrial applications that need to extend the condi-
tions accessible on Earth. The present work is a contribu-
tion to understand the stability of this important cosmic
mixture in cryogenic conditions well below 100 K and for
a pressure range going from 0 to a few kbar, therefore
including in particular conditions found in the cold inter-
stellar gas.
This work was motivated by the necessity to predict
the behaviour of this mixture in conditions that are not
easily observable yet, but that might be reached in cold
molecular clouds far from excitation sources, such as those
in outer galactic disks. Almost all earlier similar studies
about the equation of state of He and H
2
have been fo-
cusedonhightemperatureandhighpressureconditions
suited for planetary and stellar interiors (e.g., [1]).
The mutual solubility of a binary mixture with respect
to the conditions of composition, temperature, pressure
a
e-mail: yasser.safa@unige.ch
and density is also of interest for non-astrophysical appli-
cations. For example, in metallurgy a fine dispersion of
a phase during a melting process results in a significant
improvement of the mechanical properties of materials as
well as in the production of electrically and thermally well-
conducting devices.
Generally, there are two distinct classes of mixtures
according to their deviations from Raoult’s law (i.e., the
additive rule of mixing): a positive deviation corresponds
to a segregating system, and a negative deviation corre-
sponds to a short-ranged ordered alloy. The extreme de-
viations from Raoult’s law may lead to either phase sep-
aration or compound formation in the binary system. In
this work we examine such a deviation as it reflects the
energetic and structure of constituting atoms. We use the
excess Gibbs energy of mixing G
xs
M
to evaluate the degree
of segregation and the degree of miscibility of the binary
mixture.
For metallurgic applications the present work may be
helpful since, to our knowledge, very little studies have
been carried out for the treatment of liquid alloys exhibit-
ing segregation (like atoms tend to be as nearest neigh-
bors) see Singh and Sommer [2].
The idea of representing a liquid by a system of hard
spheres was originally proposed by Van Der Waals [3]; his
338 The European Physical Journal B
classical equation of state, which accounts qualitatively
for the prediction of condensation and the existence of a
liquid-vapour critical point was derived using essentially
such a simple representation. In the case of a He–H
2
mix-
ture a complication arises from the slight non-sphericity
of H
2
. In order to overcome this difficulty, following Ali
et al. [4], we make use of the shape factor for the treat-
ment of the model as a hard convex body derived from
the hard spheres system. The application of such a tool,
which is based on the scaling theory as proposed by Largo
and Solana [5], is an advanced method for dealing with
the nonsphericity of the constituents.
Until recently, most equations of state (EOS) have re-
sulted from mathematical approximations of experimen-
tal data without a more fundamental theoretical basis.
The strong point of the method adopted here is that the
relation between pressure, temperature, density and con-
centration of components is derived from the sole knowl-
edge of the intermolecular potentials. To describe the in-
termolecular repulsive and attractive interaction, we use
the double Yukawa potential (DY) which provides an ac-
curate analytical expression for the Helmholtz free energy.
When dealing with light species such as He and H
2
at low temperature, we need to take into account quan-
tum mechanical effects. Both He and H
2
have 2 protons
and 2 electrons: at first sight He appears just somewhat
heavier than H
2
. However the quantum rules and shapes
related with the electronic orbitals change completely the
macroscopic properties at low temperature. Below a few
K, H
2
can condense even at very low pressure, while He
remains fluid at normal pressure down to 0 K.
Alietal.andothershaveusedtheWigner-Kirkwood
expansion (see [4]) to take into account to first order quan-
tum effects of such a system. But after having checked and
compared with experimental thermodynamical properties,
we found that the Wigner-Kirkwood expansion diverges
at temperatures below 50 K even if we extend the quan-
tum correction to second order. To be able to reach at
least the critical point of H
2
at 33 K, we searched in the
literature for other methods and found the approach of
Royer [6] adapted to our need. Quantum contributions are
described via a renormalized Wigner-Kirkwood cumulant
expansion around 0 K, which is well adapted for our ob-
jective to describe the mixture also well below the critical
temperature, down to about the cosmic radiation back-
ground temperature of 2.73 K.
The paper is organized as follows: In this Section we
have introduced the motivations for undertaking this work
as a contribution in the study of the interstellar medium.
In Section 2, we present the hypotheses on which our
model is based, and we mention the related investigations
that we are aware of already handled by other authors. In
Section 3, we describe the intermolecular potential and the
justification of the Double Yukawa potential to describe
repulsive and attractive effects at the molecule level. The
aim of Section 4 is to introduce the formulation of the
Gibbs and Helmholtz energy via an analytical description
based on the intermolecular potential and the diameter of
hard spheres. Section 5 deals with the equation of state,
derived from the different contributions of the Helmholtz
energy. In Section 6, we treat the phase stability of the
mixture through the Gibbs energy. In Section 7,wein-
troduce the major results of this study and some related
discussion. Finally, the principal conclusions and perspec-
tives corresponding to this work are summarized in Sec-
tion 8.
2 Preliminary considerations
In this section we present the basic hypotheses for our
study. We mention several related investigations handled
by previous authors, in order to put this study in the
context of other related researches.
Our model is based on the following hypotheses:
– The inter-molecular potential model used to describe
the pairwise interaction of constituent species is the
spherically symmetric pair potential containing a short
range repulsion and a long range attraction compo-
nents.
– The model of hard convex body is used to represent
the geometry of the species in the mixture, it is de-
rived from the hard sphere system based on the scaled
particle theory SPT see [7]. A system of hard spheres
represents the simplest realistic prototype for modeling
the vapor-fluid phase separation in such a mixture.
– The mixture is considered as a pure neutral molecu-
lar phase, since we have a region of temperature T
well below 1000 K and pressure P below 1 Mbar. In
such conditions, molecular dissociation and ionization
by pressure are not expected to occur. For details on
the ionized plasma of the helium-hydrogen mixture at
high pressure, see [8]. At low temperature and mod-
erately low pressure, the transition from a molecular
phase to an atomic phase (H
2
2 H) is not expected,
further the presence of He stabilizes the molecules in
the mixture as shown in [9]. In interstellar conditions
neutrality is not always granted due to the frequent
presence of ionizing and dissociating radiations allow-
ing the coexistence of H and H
2
. However, in “dense”
molecular clouds (n>10
3
cm
−3
,3<T <50 K, still
much less dense than the best industrial vacuum) al-
most all H is converted into H
2
,sotheHe–H
2
mixture
is the relevant one there.
–ForP bellow 1 Mbar, the molecular-metallic transi-
tion is not reached. This will have an influence on the
mixing conditions, especially since He is more soluble
in H
2
than in metallic H as predicted by Stevenson
and Salpeter [10]. Results about the solubility of He
in metallic H are given by Stevenson [11], the prop-
erties of metallic H are studied under high dynamic
pressures by Nellis [12] and the details on molecular-
metallic transition of H are exposed by Chabrier [13].
– The effects of minor isotopic and trace species and
ions in astrophysical conditions (D, Li, CO, H
2
O, CH
4
,
NH
3
, . . . ) are not included in model construction.
– The gravitational separation of phases is not in-
cluded in our model. This hypothesis is justified by
assuming the gravitational field negligible, or by con-
sidering a sufficiently small region at constant pressure.
Y. Safa and D. Pfenniger: Equation of state and stability of the helium-hydrogen mixture at cryogenic temperature 339
Barotropic phenomena have been described in the
H
2
–He system in the investigations of Street [14–16].
This system belongs to an unusual class of binary
mixtures in which the more volatile component (He)
has the higher molecular weight, and at high pressure
may be more dense than the second component, even
though the former may be a gas and the latter a solid
or liquid in the pure state. As pressure passes through
a corresponding value, the liquid phase rises up and
floats on the top of the gas phase. By considering a
region around the barotropic pressure, the coexisting
phases have the same densities and the gravitational
phase separation doesn’t occur, at least for a limited
time.
– Taking into account the condition of low tempera-
ture T<50 K that we are interested in, the ortho-
para composition of H
2
is considered here to be fully
para-H
2
. This point might be improved in future
works, because the ortho-para equilibrium can be well
parametrized as a function of temperature.
3 Double Yukawa for the He–H
2
system
The estimation of the intermolecular potential energy in-
evitably involves assumptions concerning the nature of at-
traction and repulsion between molecules. Intermolecular
interaction is resulting from both short-ranged repulsion
u
HS
ij
and long-ranged attraction (or “traction”) u
t
ij
u
ij
(r)=u
HS
ij
(r)+u
t
ij
(r), (1)
while the long-ranged attraction is treated as a pertur-
bation and the short-ranged repulsion acts as an unper-
turbed reference (usually approximated by a repulsive
hard sphere).
The Lennard-Jones potential is undoubtedly the most
widely used intermolecular potential for molecular simu-
lation. It is a simple continuous potential that provides
an adequate description of intermolecular interactions for
many applications at low pressure. But the inverse-power
repulsion in LJ potential is inconsistent with quantum me-
chanical calculations and experimental data, which show
that the intermolecular repulsion has an exponential char-
acter. For this purpose the exponential-6 (α-exp-6) poten-
tial is a reasonable choice instead of the LJ potential [17].
An anomalous property of the α-exp-6 potential, how-
ever, is that at a small distance r
c
in the region of high
temperature (T>2000 K), the potential reaches a maxi-
mum value and in the limit r → 0, it diverges to −∞ [18].
As suggested in [4,19] the double Yukawa potential u
DY
may be considered as advantageous since it can fit many
other forms of empirical potentials, and, in addition, the
related integral equation of the Helmholtz free energy and
compressibility factor can be solved analytically:
u
DY
ij
=
ij
A
ij
σ
0
ij
r
e
λ
ij
(
1−r/σ
0
ij
)
− e
ν
ij
(
1−r/σ
0
ij
)
, (2)
where
ij
represents the potential depth and σ
0
ij
the posi-
tion at which the potential is zero (see Fig. 1).
Fig. 1. The double Yukawa potential for the 3 possible pair
interactions in the He–H
2
mixture.
Table 1. DY potential parameters in the He–H
2
mixture.
He–He H
2
–H
2
He–H
2
i, j 1.1 2.2 1.2
σ
0
ij
(
˙
A) 2.634 2.978 2.970
ij
/k (K) 10.57 36.40 15.50
A
ij
2.548 3.179 2.801
λ
ij
12.204 9.083 10.954
ν
ij
3.336 3.211 3.386
The terms A
ij
, λ
ij
and ν
ij
control the magnitude of
the repulsive ant attractive contributions of the double
Yukawa potential. The parameters (Tab. 1) are suitably
chosen to provides a close fit to the exp-6 potential pro-
posed in [17].
The controlling parameters (Tab. 1) are slightly non-
additive, i.e., A
12
≈ (A
11
+ A
22
)/2, λ
12
≈ (λ
11
+ λ
22
)/2
and ν
12
≈ (ν
11
+ ν
22
)/2. In contrast, the potential depth
ij
is strongly nonadditive
12
= α
√
11
22
, (3)
where the nonadditive parameter α quantifies the rela-
tive strength of the unlike pairwise interaction. In our
case (α ≈ 0.79 < 1) and the molecules are not energet-
ically alike. We are, hence, concerned with a not ather-
mal mixture (i.e., we don’t have energetically alike species
11
=
22
=
12
). In such a situation the contribution to
the free energy is predicted to be arising from both en-
thalpic and entropic effects. According to [4] the smaller
value of α (compared to 1) should drive the mixtures to-
wards demixing.
4 Gibbs energy of mixing
Considerable efforts have been spent in the recent years
to propose a fundamental physical theory describing the
340 The European Physical Journal B
reasons responsible for phase separation in a binary mix-
ture.
Thermodynamically, the Gibbs energy of mixing G
M
which depends on the enthalpy H
M
and the entropy S
M
,is
of great interest. In fact, by evaluating its deviation from
the Gibbs value of an ideal mixture G
id
, the energetic
term G
M
provides the crucial informations on the ther-
modynamic stability of the mixture. Obviously the pro-
cess could be complicated by the respective enthalpic or
entropic contributions to segregation (for details, see [2]).
The term G
M
is expressed as
G
M
= G −
i
c
i
G
0
i
, (4)
where c
i
are the mole fractions, G is the Gibbs free energy
of the mixture and G
0
i
= G(c
i
→ 1) is the free energy of
the pure constituent species i.ThevariableG relates the
pressure P to the Helmholtz free energy F
G
NkT
=
F
NkT
+
P
nkT
, (5)
where T , n
1
, N and k are respectively the temperature,
the number density, the total number of molecules, and
Boltzmann’s constant.
4.1 Hard convex body free energy
The total Helmholtz energy F for a mixture of N
moleculesisobtainedfrom
F
N
= F
id
+ F
HB
+ F
t
+ F
Q
, (6)
where F
id
is the Helmholtz energy per molecule arising
from the ideal gas mixture. It is defined with
βF
id
=
3
2
ln
h
2
2πkTm
c
1
11
m
c
2
22
+lnn +
i
c
i
ln c
i
−1, (7)
where h is the Planck’s constant, m
ii
are the atomic
masses and β is the inverse temperature β =1/kT .
The Helmholtz free energy F
HB
for the hard convex
body is given by
βF
HB
= a
mix
β
F
HS
+ F
nonadd
, (8)
where the coefficient a
mix
is the nonsphericity parameter
and will be presented in details bellow, the term F
HS
is
the Helmholtz energy of hard sphere, and F
nonadd
is the
contribution arising from the nonadditivity of the hard
sphere diameter.
The expression of F
HS
reads (see e.g., [19])
βF
HS
=
η
3
(f
1
+(2− η
3
)f
2
)
1 − η
3
+
η
3
f
3
(1 − η
3
)
2
+(f
3
+2f
2
− 1) ln(1 − η
3
), (9)
1
Note that sometimes ρ is used instead of n in the chemical-
physics literature, which leads to confusion with the elsewhere
widely adopted meaning of ρ as the mass density.
where the parameters in equation (9) are related to the
hard spheres diameters σ
ij
by
f
1
=
3y
1
y
2
y
0
y
3
, (10)
f
2
=
y
1
y
2
y
3
3
(y
4
z
1
+ y
0
z
2
), (11)
f
3
=
y
3
2
y
0
y
2
3
, (12)
y
i
=
η
i
n
, (13)
η
i
=
π
6
n(c
1
σ
i
11
+ c
2
σ
i
22
), (14)
z
1
=2c
1
c
2
σ
11
σ
22
σ
11
− σ
22
σ
11
+ σ
22
, (15)
z
2
= c
1
c
2
σ
11
σ
3
22
(σ
2
11
− σ
2
22
). (16)
The distances σ
ij
are calculated via the integration of the
correlation function
σ
ij
=
σ
0
ij
0
1 − e
−βu
ij
(r)
dr. (17)
According to [20], equation (17) may be derived from the
minimization of the free energy difference between the ref-
erence fluid (a purely short range repulsive model) and the
effective hard sphere model (including the long range at-
traction). The use of equation (17)makesσ
ij
temperature
dependent and enables us to investigate the effect of tem-
perature on G
M
and consequently the impact of tempera-
ture and pressure on the mixing conditions of binary mix-
ture (heterocoordination, segregation or phases separa-
tion). Subsequently, by taking into account the enthalpy-
entropy relation (S
M
= −∂G
M
/∂T ), the T dependence of
G
M
paves the way to illustrate the entropic contributions
of binary mixture with respect to T and P .
4.2 Non-additive free energy
The positive nonadditivity of hard sphere diameters
(σ
12
> (σ
11
+ σ
22
)/2) is predicted to cause an instabil-
ity of binary mixture as shown in many works (see for
example [21–24]). Although in the conclusion of the lat-
ter the negative nonadditivity of hard sphere diameters
(σ
12
< (σ
11
+ σ
22
)/2) is considered to not exhibit a fluid-
fluid demixing, however in [25] a demixing transition in
binary hard sphere mixture is possible for a slightly nega-
tive nonadditivity. The drawback of the major approaches
is that σ
ij
remains independent of T and hence its appli-
cability is limited.
On a more realistic basis the T dependence of σ
ij
in-
troduced via equation (17) is desirable to study the non-
additivity effect and the phase diagram of the mixture, as
shown by [19,26].
The contribution F
nonadd
is obtained by means of the
first order perturbation correction [27]
βF
nonadd
= −4πn c
1
c
2
σ
2
12
Δσ
12
g
HS
12
(σ
12
), (18)