PHASE SEPARATION OF IONIC MIXTURES

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mensionless parameters : rs, r and xl, .... nerate if the temperature is much lower ... the non-relativistic regime, and for rs < 1, .-. ... fluid is expected to crystallize.
JOURNAL DE PHYSIQUE

Colloque C2, supplément au n° 3, Tome 41, mars 1980, page C2-43

PHASE SEPARATION OF IONIC MIXTURES * J.P.

Hansen

Laboratoire de Physique Théorique des Liquides Université Pierre et Marie Curie, 4, Place Jussieu

- 7S2S0 Paris Cedex 05, France.

Résumé.- Nous p r é s e n t o n s une revue c r i t i q u e des c a l c u l s r é c e n t s du diagramme des phases de mélanges ou a l l i a g e s i o n i q u e s dans un fond c o n t i n u d ' é l e c t r o n s d é g é n é r é s . Les l i m i t e s d ' ë c r a n t a g e é l e c t r o n i q u e f a i b l e e t f o r t s o n t t r a i t é e s séparément, mais f o u r n i s s e n t des p a r a m è t r e s c r i t i q u e s s i m i l a i r e s . La v a r i a t i o n de l a t e m p é r a t u r e c r i t i q u e avec l a p r e s s i o n e t l e r a p p o r t des c h a r g e s i o n i q u e s e s t é t u d i é e e x p l i c i t e m e n t , en p a r t i c u l i e r pour des r a p p o r t s grands devant un. Parmi l e s a p p l i c a t i o n s a s t r o p h y s i q u e s , nous c o n s i d é r o n s brièvement l a p o s s i b i l i t é de l a démixtion du f e r e t de l ' h y d r o g è n e i o n i s é à l ' i n t é r i e u r du s o l e i l . Abstract.- Recent calculations of the phase diagram of binary ionic mixtures and alloys in a uniform background of degenerate electrons are critically reviewed. The limits of weak and strong electron screening are treated separately and are found to yield similar critical parameters. The dependence of the critical temperature on pressure and ionic charge ratio is explicitly investigated, in particular for ratios much larger than one. Among astrophysical applications, the possible segregation of iron from hydrogen under solar conditions is briefly considered.

1. Introduction.- Except under extreme conditions, stellar matter is made up of ions, belonging to several fully of partially ionized atomic species, and of unbound (nearly free) electrons. At sufficiently high densities and temperatures the multi-component ionic fluid can be essentially described by classical Statistical Mechanics, while the electron gas behaves like a degenerate, and possibly relativistic, Fermi gas. Due to the Pauli principle, the electronic and ionic fluids can, to a first approximation, be considered as de-coupled.

well-established that binary ionic mixtures phase separate at sufficiently low temperatures, under a wide variety of physical conditions [2j.

Some astrophysical situations where a precise knowledge of the thermodynamics and miscibility of ionic mixtures may be very important include the deep interior of the major planets, containing essentially Hydrogen and Helium ; white dwarfs and neutrons star crusts, where the highest densities of Coulombic matter are reached ; and the interior of the Sun, where the physical The present paper deals with such highly com- conditions may lead to the segregation of highly ionized Iron [3]. pressed Coulombic systems, and reviews some of the recent theoretical results which have been obtained for the thermodynamic and 2. Physical conditions and parameters.structural properties of binary (two atomic Consider a mixture of n ionic species in a species) ionic fluids. An important theoreti- volume n at a temperature T. Let N , Z and cal issue, which may have far-reaching astro- A be the number of ions, the ionic valence physical consequences, is the miscibility of and the atomic mass number,of species v two ionic species in a background of elec(1 < v < n) ; if N = Z N is the total numv v trons. Since the pioneering work of Stevenber of ions, the number density is p = N/fi, son [l] on mixtures of protons and a-partiwhile the number concentrations are x = d e s in the interior of Jupiter, it is now v N /N, and the mean ionic valence is defined Paper presented at the C.N.R.S. Intertio- as : nal Colloquium on the Physics of Dense Matter, Paris, September 1979. Equipe associee au C.N.R.S.

Due to charge neutrality, the total number of electrons is 4

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1980207

JOURNAL DE PHYSIQUE

N' = $ N ~ z\,=NZ

each species, Av, is much shorter than the ion-sphere radius, i.e. if :

and the charge density is p' = N1/3 = Z p . An equilibrium state of the mixture is characterized by the values of n + 2 independent thermodynamic variables which, for fixed N or N', can be chosen as 3, T and a set of n-1 concentrations xv. Alternarively we shall choose the following set of n + 2 dimensionless parameters : rs, r and xl, rs and r are defined as follows. The xn-l mean ion sphere radius, i.e. the radius of a sphere which, on the average, contains one ion, is related tG the number density by :

....

= a

h

The electrons, on the other hand, are degenerate if the temperature is much lower than the Fermi temperature TF, i.e. : T 20 000 K at P = 90 Mbars ! ) . Although this work yields a strong indication in favour of an enhanced miscibility gap, the results should be accepted with caution, in view of the fairly large number of poorly controlled approxi:mations which have to be mad% in this difficult calculation.

sonable to neglect electron screening completely in a first approximation. This amounts to retaining only the first term on the r.h.s. of eq,(ll) and hence to consider as a reference system a binary ionic mixture in a rigid uniform background, corresponding to the limit ce(k) = 1 for all wave numbers. We then deal with a well-defined model, which is a straightforward generalization of the widely studied "one component plasma" (OCP), or classical "Jellium" [21]. Due to the simple scale invariance of the Coulonb potential, the dimensionless excess thermodynamic properties of such a binary ionic mixture depend only on independent variables, which we choose to be x1 and. r (or equivalently T ' ) , The equation of state for this model has been accurately determined from numerical 'solQtionseofthe coupled "hypernetted chainW(HNC) integral equations for the three pair distribution functions 122,233 and "exact" Monte Carlo (MC) . simulations [23] ,' for ionic charge ratios Z2/Z1 = 2 and 3, The MC data show that the thermodynamics determined from the approximate HNC theory are very reliable, since they differ by less than 1% from the "exact" results over the whole fluid range, The main result, summarizing the extensive numerical data, is that the dimensionless ex. cess thermodynamic properties at constant temperature and charge density (i.e. at constant I?') are, to an excellent approximation, linear combinations of the corresponding pro(one component) phases. perties for the In particular the reduced excess free energy is well represented by :

=

where fo ( r ) is the reducedexcess free energy of the OCP and Tv = T' .":z The deviation from th,Ls simple linear law :

6. 5 . In the high density (rs