1
SUPPLEMENTARY INFORMATION SECTION
Equation of state for noble gases, carbon dioxide and silicate melts
For the hard sphere fluid (HS), the atoms (or molecules) are described by hard spheres of
diameter d, and its equation of state (EOS) is given with a high accuracy by the CarnahanStarling equation (Carnahan and Starling, 1969)
P  k B T
1    2   3
(1 ) 3
(S1)
where   d 3 / 6 is the packing fraction and  the number density of the fluid. Although
the HS fluid has no 
liquid-gas transition, one can show that it reproduces very well the EOS
of simple gases (noble gases and non polar molecular gases like CO2) in the one fluid region
(i.e. at supercritical temperatures) provided that a temperature dependent hard sphere
diameter, d(T), is used. In this context, it is noteworthy that the temperatures involved in the
upper mantle as well as those routinely used for solubility measurements of noble gases and
CO2 in silicate melts under pressure correspond to supercritical conditions for the volatile
species considered here (T exp ~1500-1800K as compared with T c = 150.9 K for argon, 289.7K
for xenon and 304.2K for CO 2 ). For illustration, the isothermal compressibility factor at
1600K is presented in the Supplementary Figure for He, Ne, Ar, Xe and CO 2 , respectively. In
the case of Ne, Ar and Xe the fits obtained with eqn.(S1) were found almost undistinguishable
from the data produced by the EOS of Song et Mason (Song and Mason, 1989), one of the
most reliable EOS for rare gases. For He, which is poorly known experimentally in this (P,T)
range, the fit presented here is the one corresponding to a Lennard-Jones fluid (with  / k =
10.22 and  = 2.556A, Hirshfelder et al., 1967), while for CO 2 our fit is in excellent
agreement with the accurate EOS of Pitzer and Sterner (1994).
The compressibility of silicate melts can also be described by the EOS for hard spheres,
but to account for the cohesion energy of the liquid phase it is necessary, in addition to adjust
the hard sphere diameter, to scale the EOS to a reference data point of the real material. A
convenient choice is to fit the density of the silicate melt at 1 bar for the temperature
investigated experimentally, a density value which is evaluated from the weighted sum of
partial molar volumes for the constituent oxides as given in the literature (Bottinga et al.,
1983). In the Supplementary Figure are presented the final results of the fits for silica at
2
2000K, for a MORB at 2000K and for a San Carlos olivine at 2300K. In the latter case, our
adjustments were based on the compression data of Agee and Walker (1988, 1993) for liquid
peridotite and liquid komatiite in the pressure range 0~100 kbar, two magnesian silicates of
composition closely related to that of the olivine melt. When extrapolating to olivine the
observation made by Agee and Walker (1993) that the bulk modulus is essentially invariant
with composition in these liquids (KT ~260 kbar), and evaluating the density of the olivine
melt at atmospheric pressure as described above (2.75 g/cm3 at 2300K), one is led to the
olivine compressibility curve presented in the Supplementary Figure. For the MORB
investigated here (CH98 DR11, of Jambon et al., 1986), we have fitted the shock wave
compression data obtained by Rigden et al. (1988) from an average MORB of similar
composition. In these fits, we have accounted for the difference between the isentrope
(Hugoniot) really measured and the isotherms of interest (1673-2000K in the pressure range
0-100 kbar). In the case of silica, we have used the volumetric data obtained by Gaetani et al.
(1998) in rhyolitic melts at high pressure (10-35kbar). Notice that our EOS leads to a bulk
modulus for silica of about 140 kbar at atmospheric pressure in agreement with the data of
Lange and Carmichael (1987), while at 35 kbar the bulk modulus amounts to 310 kbar, as
compared with 303  21 kbar by Gaetani et al. (1998). For the haplogranite melt considered in
the present paper, we assume that its bulk modulus is identical to the one of pure silica (which
means that the hard sphere diameters are identical for the two melts) but the reference density
at 1 bar is modified according to composition (2.27 g/cm3 for the haplogranite instead of 2.22
g/cm3 for silica). From a more general point of view, the hard sphere diameter tends to
decrease when going from highly silicic melts (e.g. d=3.27A for SiO2) to basic and ultrabasic
melts (e.g. d=3.18A for a MORB and d=2.82A for olivine). The important consequence is that
the cavities (or holes) susceptible to accommodate a solute atom of a given size occur more
frequently in the fluctuations of the liquid structure characterizing highly polymerized melts
(silicic) than in less polymerized melts (ultrabasic). This feature explains qualitatively why
the solubility of a noble gas will be greater in rhyolitic melts than in MORBs. Finally, notice
that the uncertainty in the determination of the hard sphere diameter comes mainly from the
inaccuracies in the experimental data to fit and very little from the ability of the EOS to
interpolate reliable data points. Thus an experimental uncertainty of about  10% on pressure
leads to an uncertainty of about  1% on the determination of the hard sphere diameter of the
corresponding melt.
3
Solubility of noble gases in silicate melts under pressure
As stated in the letter, to evaluate the concentration X of a noble gas in a silicate melt, it is
necessary to calculate the excess chemical potentials of the noble gas in the melt and in the
fluid phase in contact. Notice that, from the EOS described above for rare gas and CO2 fluids
and silicate melts, it is easy to deduce the respective densities of the two phases (fluid and
melt) in coexistence with each other at fixed T and P. The excess chemical potential of the
noble gas in its gaseous phase at T and P is next obtained by integration and derivation from
eqn.(S1), applying the usual thermodynamic relations (i.e.
P  (F / V ) T
and
  (F / N )T ,V ) and using for the hard sphere diameter of the noble gas, the value obtained
in the previous paragraph. As for the excess chemical potential in melt, it is evaluated
according to the scaled particle theory developed for hard spheres in the early sixties (Reiss et
al., 1960). In this theory, the excess chemical potential of a hard sphere solute (considered at
infinite dilution) in a hard sphere solvent is the work of creating a spherical cavity of desired
size in the solvent, a purely entropic contribution, given by

ex
cav
18 2 1
6
2
/ k B T = -ln(1-  ) +
(    2 ) +
(    2 )
2
1 
(1   ) 4

1   2  3
(1   ) 3
( 1  6  122  83 )
(S2)
where   d 3 / 6 is the packing fraction in the solvent of hard sphere diameter d and
number density  , and d  (d i  d ) / 2 is the radius of the cavity enclosing the solute
particle of diameter d i . In this framework, the noble gas and the melt form a true solution
(there are no well located vacancies in the solvent, the latter one accommodating the solute
through density fluctuations), but so diluted that noble gas atoms ignore each other in the
melt. Notice that e   cav is nothing but the probability of finding a cavity of a given size (di) in
ex
the pure solvent at equilibrium (i.e. when sampling all the microscopic configurations of the
solvent which realize the macroscopic state). Hence, one can conclude immediately that for a
given solvent (or melt) the larger the noble gas, the higher the entropic cost of cavity
formation and the lower the probability of absorption (solubility). The above expression is
virtually exact for hard spheres and has been very useful to interpret gas solubility data in
4
molecular and complex liquids (Pierotti, 1976). However, there is no solvation energy in a
hard sphere fluid, only the entropic contribution given above. So, the energetic contribution to
the excess chemical potential has to be evaluated independently and added to the entropic
contribution. Although the energetic contribution is small in compressed rare gas fluids and in
non-polar gases at supercritical conditions (as investigated here), this is not true for noble
gases or CO2 in a silicate melt. It has been shown (Guillot and Guissani, 1996) that the
dominant contribution to the noble gas solvation energy is the charge-induced dipole
interaction resulting from the polarization of the rare gas atom by the Coulomb field created
by the ionic species of the silicate melt. This polarization energy is attractive and is
proportional to the noble gas polarizability. Hence, its magnitude is small for a weakly
polarizable atom as helium (  He=0.2A3) and much more significant for xenon (  Xe=4.0A3),
for instance. Furthermore, it depends very little on the external pressure (due to the long range
of the coulombic interactions), contrary to the entropy of cavity formation, which varies
drastically with the density of melt (because it is a short range property). In practice, the
excess chemical potential of the noble gas in melt is composed of two terms,
ex
ex
is calculated from eqn.(S2) and where the polarization energy,
 mex   cav
 E pol , where  cav
E pol , is treated as a fitting parameter independent of pressure and adjusted to reproduce the
literature data for the solubility constant (the inverse of the Henry constant at 1 bar) of the
noble gas (or CO2 ; see below) in the silicate melt under investigation.
Elemental fractionation in MORBs
During CO 2 exsolution from a magma rising under ridges, the noble gases redistribute
between bubbles and melt according to the mass conservation law,
i

i
i
C0 = Cv + Cm
i
(S3)
3
where C 0 (in cm STP/g or in ppm) is the concentration of rare gas i in the magma before
i
i
vesiculation, C v the concentration in vesicles and C m the concentration in the vesiculated
melt. Introducing into eqn.(S3) the equilibrium conditions between the chemical potential of
the noble gas in melt and in the CO 2 phase, lead to the following expressions,
i
v
C =C
i
0
( gi /  mi )V *
1 V * ( gi /  mi ) 1



(S4)
5
i
m
C =C
1 V *
i
0
1 V * ( gi /  mi ) 1
(S5)
i
*
where V = V g /(V g +V m ) is vesicularity, and  g,m  e
i ,ex
  g,m
are the solubility parameters of
 (m) or in the CO 2 phase (g). The above relations are exact at equilibrium
noble gas i in melt
whatever
 pressure,
  and they replace the
 expressions commonly used, valid only at very low
i
i
pressure (in this case,  g  1 and  m   s k B TS i ; Henry’s law). The excess chemical potential,

 gi ,ex , of the noble gas in the CO2 phase filling the vesicles is calculated only from the cavity
term given in eqn.(S2), the polarization energy contributing practically for nothing to the

solvation of a noble gas by CO2 under pressure. As for the excess chemical potential in melt,
 mi ,ex , it is evaluated according to the procedure described in the previous paragraph.
The simplicity of the above relations is only apparent because all the quantities figuring in
*
them must be calculated self-consistently. Thus, vesicularity, V , cannot be used as a free
parameter but must be evaluated for the thermodynamic conditions prevailing at depth. If
CO 2 exsolution starts at pressure P sat , then at any pressure P<P sat during magma ascent

(considered isothermal), the mass conservation law leads to the following relationship
V*

1 V *
where  m
CO 2
=
 mCO
2
 mCO ( Psat )   s ( P)
g
2
 mCO ( P)   s ( Psat )
 (
CO
2
 1)
2

(S6)
CO
is the solubility parameter of CO 2 in melt at pressure P,  g its solubility
2

parameter
in the CO 2 phase in equilibrium (noble gases are only trace elements in this phase),
2
P,  mCO ( Psat ) the number
  CO
in melt at pressure
m (P) the number density of CO 2 molecules

2
density at the saturation pressure P sat , and  s the number density of melt (at P or P sat ). If


exsolution only occurs when some degree of CO 2 super-saturation is reached in melt, the term

-1 on the right-hand side of eqn.(S6) may be conveniently replaced by -f , where f =

N
CO2
( P) / N
CO2
eq


( P ) expresses the ratio of the number of CO 2 molecules really in melt to the
number expected at saturation.

The evaluation of vesicularity by eqn.(S6) needs to know the evolution of the concentration

of CO 2 in the silicate melt with the thermodynamic conditions. Applying our solubility
model to the dissolution of CO 2 in tholeiitic melt (then CO2 is solvated under the form of
carbonate ion and its solubility is evaluated thanks to the ansatz described in the previous
6
paragraph), one is led to a prediction which interpolates quite accurately the concentration
data of CO 2 in tholeiitic melts over the entire pressure range (0~20kbar) investigated up to
now (Pan et al., 1991; Dixon, 1997). Assuming a solubility constant of 0.44 ppm CO2/bar (a
reasonable value considering the rather large dispersion of the data at low pressure) one
obtains 535 ppm at 1 kbar, 0.75 wt% at 10 kbar and 1.6 wt% at 20 kbar, as compared with
543 ppm at 1 kbar, 0.77 wt% at 10 kbar and 1.6 wt% at 20 kbar by Pan et al (1991).
Incidentally, our model also predicts a maximum of concentration of about 2.1 wt% near 35
kbar (a feature interesting to investigate experimentally). Introducing the evolution of CO2
*
concentration with pressure into eqn.(S6) permits one to describe the evolution of V in a
tholeiitic melt during magma ascent, as illustrated in Fig.3 of the letter.
In the case where several stages of vesiculation are involved, vesicularity and concentrations
are evaluated by applying recursively equations (S4-S6) assuming complete loss of vesicles
between two events. In practice, the initial concentration of noble gas in the melt just after an
episode of vesicle loss, Cio, is identified with the concentration Cim reached in the melt just
before vesicles are lost.
Literature
Agee, C.B. & Walker, D. Static compression and olivine flotation of ultrabasic silicate liquid,
J. Geophys. Res. 93, 3437-3449 (1988)
Agee C.B. & Walker D., Olivine flotation in mantle melt, Earth and Planet. Sci. Lett. 114,
315-324 (1993)
Bottinga, Y., Richet, P. & Weill, D.F. Calculation of the density and thermal expansion
coefficient of silicate liquids, Bull. Minéral. 106, 129-138 (1983)
Carnahan, N.F. & Starling, K.E. Equation of state for nonattracting rigid spheres, J. Chem.
Phys. 51, 635-636 (1969)
Dixon J.E., Degassing of alkalic basalts, Am. Mineral. 82, 368-378 (1997)
Gaetani, G.A., Asimow, P.D. & Stolper, E. Determination of the partial molar volume of
SiO 2 in silicate liquids at elevated pressures and temperatures: a new experimental approach,
Geochim. Cosmochim. Acta 62, 2499-2508 (1998)

Guillot, B. & Guissani, Y. The solubility of rare gases in fused silica: a numerical evaluation,
J. Chem. Phys. 105, 255-270 (1996)
Hirschfelder J.O., Curtiss C.F. & Bird R.B., Molecular theory of Gases and Liquids, fourth
edition, Wiley (New York), p.1110 (1967)
7
Lange R.A. & Carmichael I.S.E., Densities of Na2O-K2O-CaO-MgO-FeO-Fe2O3-Al2O3-TiO2SiO2 liquids: New measurements and derived partial molar properties, Geochim. Cosmochim.
Acta 51, 2931-2946 (1987)
Pan, V., Holloway, J.R. & Hervig, R.L. The pressure and temperature dependence of carbon
dioxide solubility in tholeiitic basalt melts, Geochim. Cosmochim. Acta 55, 1587-1595 (1991)
Pierotti R. A. A scaled particle theory of aqueous and nonaqueous solutions, Chem. Rev. 76,
717-726 (1976)
Pitzer K.S. & Sterner M.S., Equations of state valid continuously from zero to extreme
pressures for H2O and CO2, J. Chem. Phys. 101, 3111-3116 (1994)
Reiss, H., Frisch, H.L., Helfand, E. & Lebowitz, J.L. Aspects of the statistical
thermodynamics of real fluids, J. Chem. Phys. 32, 119-124 (1960)
Rigden, S.M., Ahrens, T.J. & Stolper, E.M. Shock compression of molten silicate: Results for
a model basaltic composition, J. Geophys. Res. 93, 367-382 (1988)
Song, Y. & Mason, E.A., Statistical-mechanical theory of a new analytical equation of state,
J. Chem. Phys. 7840-7853 (1989)
8
Legend to the Supplementary Figure
Pressure (in kbar) versus density (in g/cm3) for noble gases, CO2, and various silicate melts
using the Carnahan-Starling equation of state.
Top panel. The full curves are our evaluations, the black dots are the points produced by the
accurate experiment-based EOS of Song and Mason (1989) for noble gases, and the circles
are those for CO2 as given by the EOS of Pitzer and Sterner (1994).
Bottom panel. In the case of SiO2, the full curve is our evaluation at 2000K, the squares
represent the data of Gaetani et al. (1998) for the evolution of the density between 10 and 35
kbar, while the black dot is the value of the density at one atmosphere as evaluated by Lange
and Carmichael (1987). For MORB, the full curve is our calculation at 2000K and the empty
circles are derived from the shock wave compression data of Rigden et al. (1988) for an
average MORB. Notice that, along the Hugoniot, temperature increases from 1673K at
atmospheric pressure up to 2000K at about 100 kbar (see text). For the olivine melt, the full
curve is our prediction at 2300K and the black dots represent the isothermal compressibility
of liquid komatiite at 2200K based on the data obtained by Agee and Walker (1988, 1993).
Notice that the shift between our theoretical predictions and data comes from the difference in
composition between the corresponding melts.