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Density functional theory, molecular dynamics, and differential scanningcalorimetry study of the RbFCsF phase diagram

O. Bene,1,2,a Ph. Zeller,2 M. Salanne,3,4 and R. J. M. Konings11

European Commission, Joint Research Centre, Institute for Transuranium Elements (ITU),P.O. Box 2340, 76125 Karlsruhe, Germany2

Department of Physics and Chemistry, CEA/Saclay, F-91191 Gif-sur-Yvette, France3

UPMC Univ Paris 06, UMR 7612, LI2C, F-75005 Paris, France4

CNRS, F-75005 Paris, France

Received 27 November 2008; accepted 18 February 2009; published online 7 April 2009

A multiscale modeling approach is developed to compute the phase diagram of the RbFCsF binarysystem. The mixing enthalpies of the Rb,CsF solid and liquid solutions are evaluated using densityfunctional theory and classical molecular dynamics calculations, respectively. For the solid solution,18 different configurations are studied with density functional theory and the surrounded atommodel is applied in order to compute the configurational partition function. We also measure thesolidus and liquidus equilibria using differential scanning calorimetry. Finally the RbFCsF phasediagram is constructed using the calculated excess free enthalpies of the solid and liquid solutionsand a very good agreement with our experimental data is found. 2009 American Institute ofPhysics. DOI: 10.1063/1.3097550

I. INTRODUCTION

The design of tomorrows nuclear reactors involves thedevelopment of new materials. Among them, in the moltensalt reactor MSR the fuel consists of a molten fluoridematrix in which the actinides are dissolved. Several criteriamust be fulfilled by MSR fuels, thus limiting the selectionof the salt. In the Russian concept of the nonmoderatedMSR MOSART concept1 the fuel is based on theLiFNaFBeF2 matrix with 1.3 mol % of AnF3, mostly rep-resented by 239PuF3, added as a fissile material. In a previousstudy2 five fuel alternatives based on the MOSART concepthave been presented and one of them is a system based onthe LiFNaFRbF matrix. To predict the fuel behavior interms of its melting temperature, vapor pressure, or actinidesolubility, a thermodynamic description of the systemis needed. Moreover to investigate the thermodynamicbehavior of the fuel contaminated with some amount offission products that are difficult to extract during theMSR clean-up process one must consider the presence of theCsF compound. The thermodynamic assessment of theLiFNaFRbFCsFPuF3 system has therefore been madeas described in this previous study.2 It has, however, beennoted in Ref. 3 that some additional data would be necessary

in order to describe the RbFCsF subsystem more precisely.This paper presents a thermodynamic assessment of theRbFCsF system. At the start of this work it was alreadyknown that no intermediate compound exists in this system.Both end members crystallize in the same fcc structure, thesolution is substitutional, and there is extended solubilityboth in the solid and in the liquid states. Also, the thermo-dynamic functions of both end members are well known.Hence what was missing was the exact boundaries of the

liquid+solid two-phase field, i.e., the liquidus and the soli-dus, as well as the excess contributions to the thermody-namic potentials. In this work we measure the solid-liquidequilibria using differential scanning calorimetry DSCSec. II. We also compare them to a phase diagram that webuild using the excess Gibbs energy of the Rb,CsF solidsolution and that of the Rb,CsF liquid solution, both calcu-lated here. These latter calculations involve two distinct ap-proaches, which have in common their first-principles basis,for the solid and liquid phases.

The solid solution is treated in Sec. III. First some con-figuration energies are calculated using the density functionaltheory DFT.4 Next the relevance of the bond modelBM and the surrounded atom model SAM Ref. 5 isdiscussed to extrapolate the energy to any configuration ofthe solid solution. Then we use either of two approximations,those of Bragg and Williams6 and Guggenheim,7 to computethe partition function and hence the free energy. Althoughhistorically the BraggWilliams BW and Guggenheim ap-proximations were presented within the frame of the BMwhereas the SAM was introduced combined with theGuggenheim approximation, we distinguish in the presentpaper the configurational energy models Sec. III B from theapproximations made in the calculation of Sec. III C.

The liquid solution is treated by classical molecular dy-namics MD Sec. IV with potentials built exclusively fromDFT data, as opposed to empirical or semiempirical poten-tials. This technique has already proven to be successful forthe determination of a wide range of physicochemical prop-erties of several molten salts: Dynamic diffusion coeffi-cients, electrical conductivity, viscosity8 as well as thermo-dynamic heat capacities, thermal expansions, activitycoefficients9,10 and structural quantities are well describedprovided that the interaction potentials between the involvedspecies are well chosen.aElectronic mail: [email protected].

THE JOURNAL OF CHEMICAL PHYSICS 130, 134716 2009

0021-9606/2009/13013/134716/13/$25.00 2009 American Institute of Physics130, 134716-1
http://dx.doi.org/10.1063/1.3097550http://dx.doi.org/10.1063/1.3097550http://-/?-http://-/?-http://-/?-http://dx.doi.org/10.1063/1.3097550http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://dx.doi.org/10.1063/1.3097550http://-/?-http://dx.doi.org/10.1063/1.3097550http://dx.doi.org/10.1063/1.3097550


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Many of the techniques and tools used here i.e., DSC,first-principles calculations, MD, calculation of phase dia-grams are fairly standard and have already been thoroughlydescribed elsewherewith the possible exception of our treat-ment of the solid solution. This is why more emphasis isgiven on the corresponding part of the paper Sec. III.

II. EXPERIMENTAL

The RbFCsF binary system has been investigated bySamuseva and Plyushchev,11 who used thermal analysis todetermine the solidus and liquidus points. Their reportedphase diagram, however, shows a temperature minimum onthe solidus curve without a corresponding minimum on theliquidus curve which results into a binary phase field whichis defined by two different equilibria solid solutionliquidsolution and solid solutionsolid solution, thus violating thephase rule. Therefore new measurements of the solidus andliquidus temperatures of the RbFCsF system were madein the present study. Our measurements are performedusing DSC performed on a SETARAM multidetector hightemperature calorimeter.

The samples were prepared by mixing the initial pureRbF Alfa Aesar, 99.975% metallic purity and CsFAlfa Aesar 99.99% metallic purity compounds. Thesefluorides are very hygroscopic and therefore prior to thesample preparation they were dried in a powder form undervacuum at T=453 K for 4 h. Thus obtained dry precursorswere stored in a glove box which was kept under constantargon gas flow of 6.0 grade measured long time content ofthe moisture in the glove box is 28 ppm. In order to mini-mize the weighing error, all samples were weighed directly

in the stainless steel, gas tight DSC crucibles specially de-veloped at ITU.12 Boron nitride oxygen-free, AX05 gradeliner was inserted into the crucible to avoid direct contactbetween the steel and the fluoride salts which are corrosive athigh temperatures.

Every DSC measurement consisted of three heating andcooling runs. The first run was always used to achieve goodmixing between the RbF and CsF compounds. During thisrun the salt was kept for 1 h at T=1173 K, well above themelting temperature of both components. The data from thisrun have never been considered for the analysis. The twoother runs were monitored and consisted of the heating andcooling sequences, both at temperature rate of 3 K/min. The

peak temperature was 1173 K as in the case of the prerun.Since the melting temperatures of the pure compounds werewell reproduced from the heating and cooling curves, it hasbeen assumed that there is no supercooling effect in thissystem. This observation was also supported by the fact thatRbF and CsF are highly ionic compounds and thus theircrystallization is very fast. Therefore it was possible to ana-lyze the solidus points as the onset points of the DSC peaksfrom the heating curves and the liquidus points as the onsetpoints of the DSC peaks from the cooling curves. The resultsare reported in Table I and included in Figs. 8 and 9 in Sec.V in which a comparison to the calculated phase diagrams ismade.

III. SOLID SOLUTION

We consider here a macroscopic sample made of NRbrubidium atoms, NCs cesium atoms, and N= NRb+NCs fluo-rine atoms. Molar fractions are denoted

xRbF =NRb

Nand xCsF =

NCs

N. 1

All the calculations performed in this work are done at zeropressure so that the pV term is equal to zero, enthalpy isequal to internal energy H= U, and Helmholtz energy isequal to Gibbs energy F= G. It must be noted that althoughfor ambient conditions p =1 bar, due to the fact that we aredealing with condensed phases the pV term is negligible

throughout. Statistical thermodynamics in the canonical en-semble at absolute temperature T relates the configurationalpart of the free energy to the partition function through thefollowing formulas:

F= kBT ln 2

and

= c

exp ckBT

, 3where c represents a configuration, i.e., an arrangement ofthe set of Rb and Cs atoms on the cation sublattice, c is

its energy, and kB is the Boltzmann constant. Notations arethose of Mathieu et al.5 We assume that disorder does notmix cations with anions since it is well established that an-tisite defects in these strongly ionic compounds have a verylarge positive energy. Accordingly, the configurational en-ergy is attributed to cations only and from here on we do notmention F anions any more in this section. In contrast to aconfiguration, a solid solution is a macroscopic conceptencompassing all these microscopic configurations. Theweight of a configuration in a given solid solution dependson external constraints as described by thermodynamic vari-ables e.g., p, T, U, S, xRbF, xCsF, Rb, and Cs, out of whichwe choose here p =0, T, and xRbF. Mixing quantities obey

TABLE I. Solidus and liquidus temperatures of the RbFCsF system ob-tained by DSC in this study.

xCsF

TsolidusK

TliquidusK

0 1060.90.069 1022.6 1040.60.088 958.9 9780.141 1015.6 1033.1

0.236 985.9 1010.70.297 973.7 999.90.396 961.7 983.80.506 962.1 978.20.597 947.1 967.90.694 953.1 961.30.788 941.4 953.20.839 944.4 954.60.858 960.3 999.41 959.4

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similar relationships as shown in Eqs. 2 and 3. In thisstudy it is assumed that configurational energies do not de-pend on temperature and they are computed for T=0 K. Inother words the vibrational contributions to the excess freeenergy are neglected. This includes the so-called zero-pointenergy, i.e., the phonon energy at T=0 K, since we do notcompute any phonon spectrum.

A. DFT calculation

1. CASTEP parameters

Our calculations are performed using the plane-wave pseudopotential code CASTEP which implementsDFT and has been extensively described elsewhere.13 Weuse the generalized gradient approximation of Perdew-Burke-Ernzerhof GGA-PBE approximation14 for the ex-change correlation functional. The first Brillouin zone issampled with the MonkhorstPack scheme15 with intervalsno larger than 0.040 1 on all axes. We use the MATERIALSSTUDIO Ref. 16 interface for launching the calculation andanalyzing the results as well as various interfaces and toolswithin the CYGWIN/X package.17 The pseudopotentials arealso taken from the MATERIALS STUDIO library. To identifythe best cut-off energy value, several calculations of tetrag-onal RbCsF2 P

4m

mm space group have been performedwith cut-off energies ranging from 350 to 500 eV by 50 eVsteps. A satisfactory accuracy is achieved for a cut-off energyof 400 eV and this value has been selected for all our calcu-lations. All elements are described using Vanderbilt schemeultrasoft pseudopotentials18 with configurations He2s22p5

for F, Ca 3d104s24p65s1 for Rb, and Sr 4d105s25p66s1 forCs. For each configuration a geometry optimization is per-formed, i.e., ionic positions and cell parameters are varied so

as to minimize the total energy.Pseudopotentials from the MATERIALS STUDIO library areprovided with several convergence and validity tests. In or-der to assess the validity and the performance of our chosenset of parametersincluding these pseudopotentialswe havecomplemented those tests by computing equilibrium geom-etries and atomization energies of pure Rb, Cs, RbF, and CsFcrystals and comparing them to experimental values takenfrom the literature.19,20 For atomization energies we have in-cluded in the calculated data the zero-point energies com-puted with CASTEP all of them in the 1 kJ/mol atom order ofmagnitude and in the experimental data all the correctionsdescribing the transition from the standard state at 298 K to

the zero pressure and zero temperature conditions. The re-sults are summarized in Table II. All lattice parameters agreeto within 1%, which is a good result consistentwith CASTEP standard performances for these settings,21 con-firming the quality of the pseudopotentials. All calculatedatomization energies happen to be underestimated by71 kJ/mol atom. This similarity is of course fortuitous,and the relative differences are 10% for the pure metals and2% for RbF and CsF. The discrepancy for the pure metals isin the usual range, but it may also reflect the fact that forthose simple metals the GGA usually performs worse thanthe local density approximation. Since this paper focuses onionic and partly covalent materials GGA is arguably the best

choice. For RbF and CsF these energy differences are accept-able given the purpose of this work. In fact although errorson energies of the order of 7 kJ/mol atom would in generalpreclude the computation of a phase diagram completelyab initio, relative errors of a few percentage on the excessquantities are quite satisfactory and we may in fact expect

small relative errors on the excess quantities due to well-known error cancellation properties in first-principlescalculations.4,21

2. Excess configurational energies

The energies of 18 different configurations of theRb,CsF solid solution have been calculated using CASTEP.Energies of the pure RbF and CsF end members have thenbeen proportionally subtracted from the energy of eachRb,CsF solid solution configuration in order to obtain theexcess configurational energy as

xsRbxCsyFx+y =RbxCsyFx+y xRbF

yCsF. 4

The choice of the various configurations was made by arbi-trary displacement of the Rb and Cs atoms on the cationsublattice in order to cover a large array of nearest neighborenvironments. The calculated excess configurational energiesare shown in Fig. 1 and reported in Table III. Although someof the configurations have the same crystallographic spacegroup and contain the same amounts of Rb and Cs cationsper unit cell e.g., case of configurations 6, 8, and 9 thearrangements of the cations within the cation sublattice aredifferent, as well as their cell parameters, so that they corre-spond to different configurations and are in general associ-

ated with different energies.

B. Configurational energy models

In order to extrapolate see Eq. 3 from our CASTEPdata set to any configuration, several models can be consid-ered. A very common one, used by Guggenheim in his treat-ment of mixtures,7 can be called the bond model BM. Itdistributes the configurational energy among the first neigh-bor bonds, that is, in the present case, among the cation-cation bonds since we neglect F anions. Three parametersare introduced: RbRb, CsCs, and RbCs =CsRb which describethe bond energies according to their type. The energy of

TABLE II. Atomization energies and lattice parameters of some chosenreference compounds.

Compound

Lattice parametera

Atomization energyb

kJ mol1

Experimentalc CASTEP Experimentald CASTEP

Rb metal 5.70 5.70 82.192 75.4Cs metal 6.14 6.14 78.014 70.0

RbF 5.642 5.710 718.180 704.5CsF 6.03 6.100 709.598 693.0

aAll these compounds crystallize in a cubic structure.bAt 0 K and 0 bar.cReference 20.dReferences 19 and 42.

134716-3 Investigation of the RbFCsF phase diagram J. Chem. Phys. 130, 134716 2009
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configuration c and the associated excess energy are

c =z

2NRbRbRb + NCsCsCs + W

5and xsc = W,

where z is the lattice dependent coordination parameterwhich in case of the Rb,CsF fcc lattice is equal to 12, W isthe number of RbCs nearest neighbor pairs in the sample,and , the single parameter of the model for excess energy,has the dimension of an energy and is defined as

= RbCs RbRb + CsCs

2. 6

The BM cannot be satisfactorily fitted to our CASTEP data setas it leads to a rms deviation of the order of the excessenergies themselves. Therefore we switched to the moreelaborate treatment provided by the SAM.

The SAM has been presented by Mathieu et al.5,22 todescribe binary alloys. We apply it here to the pseudobinarymixture by treating RbF and CsF as two atomic species. Inthe following we briefly summarize the original derivationby Mathieu et al. using as much as possible the notations of

their work.5

The main characteristic of the SAM is the el-ementary configurational support as a central atom in ourcase a cation in the force field of its z nearest neighbors onthe cation sublattice. This replaces the partitioning based onpair interactions used in the BM described above. A configu-ration is characterized by a set of 2z +1=26 numbers de-noted M0 , M1 , . . . , M12 , P0 , P1 , . . . P12 where Mj is the num-ber of Rb atoms having exactly j Cs first neighbors and Pi isthe number of Cs atoms having exactly i Rb first neighbors.The corresponding individual atom energies are denoted ERb

j

and ECsi according to the atom type and its surroundings. The

excess configurational energy of configuration c is thereforesimply equal to

SAMxs c =SAM

xs Mj,Pi

=SAMxs M1, .. . ,M12, P1, .. . , P12

= j=1

12

MjERbj ERb

0 + i=1

12

PiECsi ECs

0 . 7

Fitting Eq. 7 to CASTEP data requires optimization of2z =24 parameters 12 ERb

j ERb0 and 12 ECs

i ECs0 excess en-

ergies. Beyond the fact that this is impossible based on theknowledge of only 18 configurations calculated in thiswork, it would not make much physical sense to fit theseparameters completely independently of one another by re-curring to, say, 26 configurations instead of 18. Nevertheless,in the same way as in the original work by Mathieu et al.,22

we observe that a three-parameter law of variation of theseexcess energies as a function of the atom surroundings issufficient in order to obtain a good agreement between theSAM and CASTEP results. Introducing the molar excessenergies

Uj = NAERbj ERb

0 and Vi = NAECsi ECs

0 , 8

where NA is Avogadros number, the parabolic laws used inthis study are

Uj = 1j + 2j2 9

and

Vi = 1i + 2i2, 10

where 1 =0 Ref. 23 and 2, 1, and 2 are the three pa-rameters optimized by least squares fit to

2 = + 76.4 J/

mol cation, 11

1 = 550.9 J/mol cation, 12

2 = + 85.2 J/mol cation. 13

Excess configurational energies obtained from the SAMare plotted in Fig. 1 together with CASTEP values and avery good data reproduction is evident. These results are alsoreported in Table III as well as the difference betweenCASTEP and the SAM giving a maximum absolute error of0.141 J/mol.

We conclude this section with a few relationships5 which

will be needed in the next section.

j=0

12

Mj = NRb, j=0

12

Pj = NCs 14

and

j=0

12

jMj = j=0

12

jP j = W, 15

where W is the number of RbCs nearest neighbor pairs, arederived from obvious conservation laws.

FIG. 1. Excess configurational energy xs of the various configurations ofthe Rb,CsF solid solution calculated with CASTEP symbol and with the SAM symbol, Sec. III B. Solid curve Excessenthalpy Hxs of the solid solution obtained with the BW thermodynamicsmodel applied to the SAM data Sec. III C 1.

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m j =Mj

NCzj

and pi =Pi

NCzi, 16

where Csr= sr= s! / r!s r! is the usual binomial coeffi-

cient, define useful auxiliary variables m j and pi.

C. Calculation of the partition function

At this point all the terms of the sum in Eq. 3 areknown, at least formally, from Eqs. 713. To calculate thesum two basic models may be applied, the BW model6 andthe Guggenheim model7 also known as the quasichemicalmodel.

We have just seen that the configurations can be sortedusing an auxiliary parameter, say, A, such that all configura-tions associated with A = a have the same energy Ea. Thenumber of configurations having A = a is the degeneracy de-noted ga. In the BM A is simply equal to W, the number of

RbCs nearest neighbors pairs, while in the SAM A is anarray of 26 numbers M0 , M1 , . . . , M12 , P0 , P1 , . . . , P12 seeEq. 7. What matters here is that any single configuration ccan be given a specific value a = Ac, be it in the BM or inthe SAM. After having grouped all the configurations whichhave the same value of A, and therefore the same energydenoted from now on a, Eq. 3 becomes

= a=all A

gaexp akBT

17

and the total number of configurations is

N!

NRb ! NCs!=

a=all Aga. 18

Given that ga is always positive, there exists a value a suchthat Eqs. 17 and 18 combine into

=

N!

NRb ! NCs! expa

kBT , 19where a is in general a function of T, NRb, and NCs and =a is a weighed average of the configurational energy.The free energy of mixing is obtained from using Eq. 2

Fm =a + kBTNRb ln xRbF + NCsF ln xCsF, 20

and the Boltzmann law for the probability of an arbitraryconfiguration c to occur becomes

probc =NRb ! NCs!

N!expa Ac

kBT . 21

1. BraggWilliams model

The BW model considers a completely disordered solu-tion. From the point of view of configurational arrangementsthis assumption implies that the probability for any givencation site to be occupied by a Cs resp. Rb cation is equalto xCsF resp. xRbF whatever its surroundings. Therefore theprobability for a given Rb atom to have exactly j Cs atomsand z j Rb atoms among its 12 nearest neighbors is givenby the binomial distribution as

TABLE III. Calculated excess configurational energy of some configurations of the Rb,CsF solid solution.

No. Compositiona Structure Space group IT number CASTEPxs b

SAMxs b,c SAM

xs CASTEPxs b,c

1 RbF Cubic Fm3m 225 0 0 0

2 Rb7CsF8 Cubic Fm3m 225 0.935 0.937 0.0023 Rb7CsF8 Tetragonal P

4m

mm 123 1.154 1.013 0.1414 Rb3CsF4 Orthorhombic Cmmm 65 1.401 1.441 0.040

5 Rb5Cs3F8 Monoclinic P2m

10 2.207 2.107 0.100

6 Rb20Cs12F32 Monoclinic Pm 6 2.263 2.199 0.0647 Rb20Cs12F32 Monoclinic P

2m

10 1.499 1.635 0.1368 Rb20Cs12F32 Monoclinic Pm 6 1.816 1.877 0.0619 Rb20Cs12F32 Monoclinic Pm 6 2.319 2.431 0.112

10 RbCsF2 Tetragonal P4m

mm 123 2.946 2.968 0.02211 Rb2Cs2F4 Orthorhombic Pmmn 59 1.319 1.256 0.062

12 Rb3Cs3F6 Hexagonal R3m 166 1.256 1.256 0.00013 Rb4Cs4F8 Tetragonal Amm2 38 2.142 2.112 0.03014 Rb4Cs4F8 Tetragonal Pmma 51 2.142 2.112 0.03015 Rb12Cs20F32 Monoclinic Pm 6 1.663 1.654 0.010

16 Rb3Cs5F8 Monoclinic P2m

10 2.081 2.030 0.05017 RbCs3F4 Orthorhombic Cmmm 65 1.270 1.299 0.029

18 RbCs7F8 Cubic Fm3m 225 0.796 0.804 0.00919 RbCs7F8 Tetragonal P

4m

mm 123 0.845 0.890 0.044

20 CsF Cubic Fm3m 225 0 0 0

aThe indicated composition corresponds to the number of atoms per unit cell.bAll units are in kJ/mol of one RbxCs1xF unit formula.cSAM stands for surrounded atom model Sec. III B.

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Mj

NRb= Cz

jxCsF

jxRbF

zj , 22

from which we get

m j = xCsF

jxRbF

zj+1 and symmetrically p j = xRbF

jxCsF

zj+1 .

23

In this and the following sections all SAM quantities relative

to the BW solution are denoted with a star superscript. Theseexpressions together with Eq. 15 yield the number of near-est neighbor RbCs bonds,

W = NzxCsFxRbF. 24

From a thermodynamics point of view the BW assumption ofcomplete disorder implies that the entropy of mixing is equalto that of ideal mixing,

SBWm = NkBxRbF ln xRbF + xCsF ln xCsF . 25

A comparison of this expression with Eq. 20 shows imme-diately that the internal energy of mixing is

UBWm

= FBWm

+ TSBWm

=a

26and due to the general law U/Tp=0,xCsF= TS /Tp=0,xCsFit appears that a, like SBW

m , is temperature independent.Evaluating it as an average configurational energy in thelimit of infinite temperature so that all configurations havethe same probability of occurrence, using the linearity ofEq. 7 with respect to Mj and Pi and using the fact that Mj

and Pi given by Eqs. 22 and 23 are also configurational

averages, we obtain

UBWm = lim

x

1

all c

c = j=1

12

MjUj + P j

Vj =a .

27This derivation, in the course of which all exponential Bolt-zmann terms of Eq. 21 are taken equal to 1, clearly showswhere the approximation lies and that the BW model is onlyjustified at high enough temperatures. Using Eqs. 23 and913 and summing over index j leads to the followingexpression for the molar excess energy and enthalpy equalin the present case:

Hm,BWxs Um,BW

xs = xRbF1 xRbFL11 + L21xRbF, 28

where

L11 = z1 + 1 + z2 + 2 = 5413 J/mol 29

and

L21 = zz 12 2 = 1162 J/mol. 30

The resulting molar excess enthalpy is shown in Fig. 1together with the calculated molar configurational en-ergies. The peak of the curve corresponds toHm,BW

xs =1.502 kJ/mol and xCsF=0.48 so that a slight asym-metry toward the RbF side is found. As shown by Mathieuet al.

22 ifUj and Vi in Eqs. 9 and 10 followed linear laws2 =2 =0 the SAM would be completely equivalent to theBM with = 1 +1 /NA and it would describe a symmetricregular solution. Since a quadratic law is obeyed there is a

clear departure from the BM and the solution is asymmetric.This has to do with the fact that the substitution of a Cs atominto a RbF crystal is energetically different from the oppositecase of substituting a Rb atom into a CsF crystal. The maincharacteristics of the BW model are the temperature indepen-dence of Hxs and the absence of any excess entropy. Its as-sumption of complete disorder is best justified in the limitingcase of infinite temperature. It is interesting to note that as

illustrated by Fig. 1 any single one among the configurationsstudied with CASTEP would in itself generally give a poorestimate of the excess internal energy/enthalpy of the solu-tion. In fact even a zero excess configurational energy ispossible for any concentrationalthough it is not displayedon the graph. This zero energy is obtained for a single con-figuration, that with complete spatial separation of both com-ponents. The excess internal energy of the solid solution, as athermodynamic potential, takes into account the fact that inthe real crystal disorder may favor higher configurational en-ergies due to their higher statistical weight.

2. The quasichemical Guggenheim model

Equation 21 shows that if two configurations have alarge energy difference the one with the lowest energy has asignificantly larger probability of occurrence. This fact is ne-glected by the BW model but it is taken into account by theGuggenheim model which we now briefly recall. A crucialstep in this model is to write ga as

ga = hNRb,NCsua , 31

where ua is some known expression which will be madeexplicit later that makes up a fair approximation to g butalso needs to be complemented by h, called the normaliza-tion factor, which only depends on NRb and NCs and is notknown explicitly. The Guggenheim approximation consistsin making use of the theorem of the dominant distributionwhich states that the sum in Eq. 17 Eq. 18 can be re-placed by its maximum term. Forgetting momentarily thatthe starred quantities have already been defined above withinthe frame of the BW model, we define here a such thatga is the maximum term of the sum in Eq. 18. We thusobtain a, which is a function of NRb and NCs but is indepen-dent of temperature, by solving

duda

a=a= 0. 32

Equation 18 becomes

N!

NRb ! NCs!= ga, 33

from which we get hNRb , NCs as

hNRb,NCs =N!

NRb ! NCs!

1

ua. 34

In a similar way the value of a that maximizes the runningterm in Eq. 17, denoted a, is obtained as a function of NRb,NCs, and temperature T by solving

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7/14

dda

uaexp akBT

a=a

= 0. 35

The Guggenheim approximation leads to

= gaexp akBT

36

and using Eq. 34 to eliminate hNRb, NCs we obtain thefinal expression of as

=N!

NRb ! NCs!

ua

uaexp a

kBT . 37

The free energy of mixing is obtained from using Eq. 2

Fm =a + kBTNRb ln xRbF + NCsF ln xCsF ln uaua

.38

As shown in the original paper5 the following treacherously

simple expressions for the molar excess internal energy,enthalpy, and entropy do hold:24

Hxs Uxs =NA

Na =

j=1

12

CzjmjUj + pjVj 39

and

Sxs =R

Nln

ua

ua, 40

where R is the perfect gas constant. Let us now specify ex-

plicitly function ua.In the BM, which we recall for illustration purposes, a isthe number W of RbCs nearest neighbor pairs in the sampleand ua is defined as

ua = z

2N!

z2

NRb a

2 ! z

2NCs

a

2 ! a

2 ! a

2!

. 41

Therefore a, determined by du /da =0 using Stirlings for-mula, is equal to

a

= W

= N z xRbF xCsF 42

and a is the solution of Eq. 35 which reads

a2

zNRb azNCs a= exp 2

kBT 43

and gives its name to the quasichemical model due to theresemblance with the mass-action law. However, as stated inSec. III B, in the present case the BM is not satisfactory.

In the SAM the expressions are much more complicateddue to the fact that a is an array of 26 parameters. Functionu is equal to

u =NRb ! NCs!

Nm0 ! Nm1!z

Nm j!Cz

j

Np0 ! Npi!Cz

i

44

and lnu is well approximated by

lnu = Nj=0

12

Czjmj ln m j + pj ln pj. 45

The starred quantities a, i.e., the set of Mj and Pi

thatcancel the differential du in Eq. 32 end up having the sameexpressions as those already given within the BW modelEqs. 22 and 23: Maximization of g in Eq. 18 isequivalent to the hypothesis of total disorder and there is noambiguity in our notations. The barred quantities are equalto25

mj = m0ej exp Uj

RT , 46

pi =

p0ei

expVi

RT , 47where

m0 =xRbF

Z, 48

p0 =1 xRbF

Q, 49

Z = 0

z

Czjej exp Uj

RT , 50

Q = 0

z

Cziei exp Vi

RT , 51

and is such that e is the single real positive root of thepolynomial of degree 2z=24 in e defined by

Re = ezxRbFQZ

+ 1 xRbFZQ

. 52

These equations, which complete the presentation of theGuggenheim approximation applied to the SAM, have nosimple analytical solution in the general case. A numericalsolution is conveniently obtained using SCILAB.26

The BW model of Sec. III C 1 can be derived using thepresent Guggenheim method by just replacing the definitionofain Eq. 35 by a= a, where a is given by Eq. 32. ThusEqs. 37 and 38 become

BW =N!

NRb ! NCs!exp a

kBT 53

and

FBWm =a + NkBTxRbF ln xRbF + xCsF ln xCsF , 54

from which Eqs. 27 and 25 can be deduced. This deriva-tion clearly shows why the BW model is also called zero-

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order approximation while the Guggenheim model is called

first order approximation. It also shows that the differencebetween both models can be quantified by aa.

Going back to the approach behind Eq. 31, a carefulexamination of Eq. 41 reveals that ua does not qualify asthe number of configurations having a number of RbCsbonds equal to a because of interferences between the differ-ent types of pairs and of double counting of oriented RbCsbonds. Similarly ua in Eq. 44 is not quite equal to thenumber of configurations having A =a because a given atombelongs to the surroundings of several other atoms and thiskind of interference is not taken into account.27 Therefore thesum ofu over all values ofa allowed by the model is neverexactly equal to the total number of configurations. All theseeffects would be very difficult to take into account properly;hence the correction introduced with function h partlyremoves these defects and leads to the correct orders ofmagnitude.

The excess Gibbs energy as a function of T and xCsF isreported in Fig. 2 where a comparison between the Guggen-heim and the BW models is made.

The solid solution is described in the present model us-ing the three fitting parameters denoted 2, 1, and 2 andgiven by Eqs. 1113. A more physically intuitive interpre-tation of these three parameters will now be given by resort-ing to three relationships given in Refs. 5 and 22. Two of

them pertain to the excess partial molar enthalpy of CsF in arubidium-rich solution xRbF close to 1, approximately givenby the following limiting value:

Csxs, = Vz + zU1 = z1 + 1 + 2 + z2, 55

and symmetrically to the excess partial molar enthalpy ofRbF in a cesium-rich solution

Rbxs, = Uz + zV1 = z1 + 1 + z2 + 2. 56

In the present case we find Csxs,=5417 J /mol and

Rbxs,=6580 J /mol. These values and the expressions for

these limiting partial molar enthalpies are valid for both theBW and the Guggenheim model: They are insensitive to tem-

perature. The third relationship, written here in the limitingcase of a cesium-rich solution, reads

SRO = 1 P2

P2

= 1 exp V2 2V1RT

= 1 exp 22RT

57

and defines a parameter that we call SRO for short-rangeorder. Recalling that P2 is the proportion of cesium atomsthat have two rubidium first neighbors, SRO quantifies a dif-ference between the BW P

2

and the Guggenheim modelsP2 in terms of cation surroundings in the first coordinationshell, i.e., in fact some kind of short-range order associatedwith the Guggenheim model since the BW model is based onan assumption of complete disorder. Now by inverting Eqs.5557 the individual excess cation energies Uj and Vi interms of which the SAM describes the solid solution can beexpressed using Rb

xs,, Csxs,, and SRO. For given Rb

xs, andCs

xs, the BW model is insensitive to the third parameter,SRO, whereas the results of the Guggenheim model dependon it. This is illustrated in Fig. 3 where the excess enthalpy isplotted for Cs

xs, and Rbxs, fixed at the values cited above for

a constant temperature T=1050 K and for a range of SRO

values. By construction all the curves merge at the endpoints xRbF0 and xRbF 1 and they mostly differ in themiddle range of xCsF. This figure also shows that accordingto CASTEP and the SAM, i.e., 2 given by Eq. 13 orSRO=0.019, short-range ordering occurs indeed in theRb,CsF solid solution but its energetic influence remainsquite limited.

Let us now relate this to a more commonly used defini-tion of short-range order. In a given configuration the aver-age surroundings of a Rb cation, i.e., the average number ofCs cations around it, is equal to j=0

z jMj / j=0z Mj. Hence,

given a Rb cation and a given site s among its nearest neigh-bors, the probability probCs Rb that site s is occupied by a

FIG. 2. Color online Excess Gibbs energy of the Rb,CsF solid solution.The solid curves, obtained with the Guggenheim approximation, are iso-therms between 300 and 900 K with a 100 K step. The dashed line isobtained with the BW approximation.

FIG. 3. Color online Excess enthalpy of the solid solution calculated fordifferent values of SRO with fixed T=1050 K, Cs

xs,=5417 J /mol, andRb

xs,=6580 J /mol. Dashed SRO0 and solid lines SRO0 areobtained with the Guggenheim approximation and the thick lineSRO=0.019 corresponds to CASTEP results. The dotted line is obtainedwith the BW approximation and is insensitive to SRO.

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Cs cation is equal to 1 / zNRb j=0z jMj and one wants to com-

pare it to xCsF. This suggests to define a short-range orderparameter by

= 1 probCsRb

xCsF= 1

1

NzxRbFxCsFj=0

z

jMj = 1 W

W,

58

where W is obtained from Eqs. 15, 16, 46, and 47 andcomputed numerically together with the solution of theGuggenheim model. This is in fact the definition of the

Cowley order parameter for the first coordination sphere.

28

Ifthe solution is completely disordered then =0. If0 thesolution shows a trend to intermediate compound formation.If0 the trend is toward segregation or phase separation.Here is a function ofT and xCsF and it also depends on theSAM parameters Cs

xs,, Rbxs,, and SRO. In Fig. 4 we plot

versus SRO for several values ofxRbF while Csxs,, Rb

xs,, andT remain fixed at the same values as for Fig. 3. The figureshows that SRO is in good qualitative agreement with themore standard Cowley definition and that it has the sameorder of magnitude. A plot of versus SRO for a differenttemperature would also yield an almost linear dependence,only with a different slope. It is interesting to note that SRO

is explicitly based on correlations in the first coordinationsphere whereas only reflects the average surroundings, andthis difference appears most strikingly in highly diluted so-lutions where becomes worthless since at first order sucha solution is always completely disordered. Besides, the factthat and SRO are so closely linked is due to the shapefunctions we chose for the surrounding atom energies Eqs.9 and 10, whereas with arbitrary Uj and Vi there wouldnot necessarily be such similarity between them.29 The bigmerit of SRO is that it provides us with a clear and simplephysical interpretation for the third parameter of the SAM,whereas the Guggenheim model, and therefore W or aswell, has no analytical solution in the general case.

IV. LIQUID SOLUTION: MOLECULAR DYNAMICS

The Rb,CsF liquid solution has been studied by MDsimulations. In the case of molten fluorides, the potential isbest described as the sum of four different components:Charge-charge, dispersion, overlap repulsion, andpolarization.30 The first three components are purely pairwiseadditive; first the charge-charge term is

Vqq = ij

qiqj

rij, 59

where qi is the charge on ion i, and formal charges are usedthroughout. The dispersion component includes dipole-dipole and dipole-quadrupole terms,

Vdisp = ij

1 f6bijrij C6,ijrij6 + 1 f8bijrijC8,ij

rij8 ,60

where C6,ij C8,ij is the dipole-dipole dipole-quadrupoledispersion coefficient, and 1 fn are TangToennies disper-

sion damping functions31 describing the short-range penetra-tion correction to the asymptotic multipole expansion ofdispersion32 fn0 =1 and fn =0. These functions takethe form

fnx = ex

k=0

nxk

k!61

and the parameter bij represents the distance at which thecorrection begins to be taken into account. The third term ofthe interaction potential, the repulsion overlap component, isgiven by

Vrep

= ij Bijeaij rij

. 62

The polarization part of the potential includes charge-dipole and dipole-dipole terms

Vpol = i,j

qi,j1 cji f4bD,ijrijT1

+ i,j

q j,i1 cij f4bD,ijrijT1

i,j

,i,jT2 +

i

1

2ii

2. 63

Here T1

and T2

are the charge-dipole and dipole-dipoleinteraction tensors while i is the polarizability of ion i.Again, TangToennies functions are included to account forthe short-range effects. The set of induced dipoles iiN istreated as 3N additional degrees of freedom of the system.The dipoles are determined at each time step by minimiza-tion of the total polarization energy and they depend on thepositions of all the atoms at the corresponding time; there-fore the polarization part of the potential is considered to bea many body term.

All the parameters necessary to simulate RbFCsF mix-tures have been determined from a recently developed first-principles procedure.33,34 The pair parameters are summa-

FIG. 4. Color online The correlation between short-range order parametersSRO and at different values of xRbF. All curves are calculated with fixedT=1050 K, Cs

xs,=5417 J /mol, and Rbxs,=6580 J /mol.

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rized in Table IV. The polarizabilities were, respectively, of7.9, 8.4, and 14.8 a.u. for F, Rb+, and Cs+ ions.

The MD simulations were performed on 11 molten saltcompositions ranging from pure RbF to pure CsF. All thecorresponding simulation cells contained 432 ionic pairs.The mixtures were simulated in the NPT ensemble followingthe method described by Martyna et al.,35 with a pressurefixed at 0 GPa and a temperature of 1200 K. We chose a timestep of 0.5 fs and after 100 ps of equilibration, productionruns of 200 ps were conducted for each composition.

Enthalpy of the systems was sampled each 50 fs duringthe production runs. This quantity was then averaged overthe full simulation, and its composition dependence is dis-played in Fig. 5. The error bars were determined by blockaveraging. We could then extract the enthalpy of mixing foreach composition, which corresponds to the difference be-tween the obtained enthalpy and the value corresponding toideal mixing between the two components according toEq. 4.

The variation of the enthalpy of mixing with composi-tion is displayed in Fig. 6. It does not exceed 140 J mol1,which is a very small value in this type of system. The errorbars have the same amplitude as in Fig. 5, but because of the

small quantities involved, they now have the same order ofmagnitude as the enthalpies of mixing themselves. As a com-parison, in molten fluoride systems which involve the mixingof very different cations, such as Li+ and Cs+ cations, theexcess enthalpy reaches a value of 3330 J mol1, with asimilar error bar and at the same temperature, as calculatedwithin the frame of this study. This means that the RbF andCsF systems can be considered to mix almost ideally.

The mixing properties in a liquid depend a lot on the

structural properties of its various components.36,37 In puremolten salts, the first-shell structure of a given cation varieswith its size and valence charge. Longer range structure ismainly understood on the basis of packing and Coulombordering. When adding a second component, this structure isbroken to an extent which depends on the compatibility ofthe two materials. For example, one can distinguish the net-work formers such as BeF2 Refs. 8 and 38 from the net-work breakers such as CsF. In the case of the RbFCsF mix-ture, the two cations have some very similar first-shell

structures. In Fig. 7, we have plotted the RbF, CsF, andFF radial distribution functions for three different composi-tions. First, it appears that the first maxima of the cation-anion functions correspond to distances of, respectively, 2.6and 2.7 for Rb+ and Cs+. Second, all the radial distributionfunctions change only slightly upon mixing, which meansthat all the species have similar environments in the pureliquids and in the mixture. This is of course consistent withthe variations obtained for the enthalpies of mixing, and itconfirms the nearly ideal behavior of the mixture.

V. THERMODYNAMIC ASSESSMENT OF THE RbFCsFPHASE DIAGRAM

As described above, two models, the BW and theGuggenheim models, have been used to interpret the DFTdata relative to the solid solution. In order to simplify and tokeep a compatibility with our developed database,2 the re-sults from these models have not been directly used for theassessment of the RbFCsF phase diagram. For the excess

TABLE IV. Parameters of the interaction potential a.u..

Ion pair ij Bij aij C6,ij C8,ij bij bD,ij cij c ji

F F 282.3 2.444 15.0 150.0 1.9 1.0 0.0 0.0F Rb+ 150.96 1.961 0.001 0.001 1.9 1.822 3.463 0.440F Cs+ 151.12 1.874 10.95 109.5 1.9 1.930 3.391 0.485Rb+Rb+ 1.0 5.0 0.001 0.001 1.9 1.0 0.0 0.0Rb+Cs+ 5000.0 3.0 0.001 0.001 1.9 1.0 0.0 0.0Cs+ Cs+ 5000.0 3.0 8.0 80.0 1.9 1.0 0.0 0.0

FIG. 5. Computed mean enthalpies for all the compositions.

FIG. 6. Enthalpy of mixing and their error bars for all consideredcompositions calculated for T=1200 K. Solid line Fit using the modifiedquasichemical model Sec. V.

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Gibbs energy description of the Rb,CsF solid solution theclassical polynomial formalism generally defined as

Gxs = i,j=1

n

x1i x2

j Lij 64

has been used, where Lij are the parameters to be optimizedand are written in a polynomial form. In the case of the BWresults the expression is exact and the Lij coefficients wereobtained analytically according to Eqs. 2830 using the2, 1, and 2 parameters from Eqs. 1113 and the cor-responding excess Gibbs energy is

Gm,BWxs = xRbF1 xRbF5413

+ xRbF2 1 xRbF1162 J mol

1 . 65

The excess Gibbs energy function based on the Guggenheim

model was obtained by the least squares fit of its results andthe final equation, valid for the temperature range 3001200K, is given below

Gm,Gugxs = xRbF1 xRbF5423 544 000

TK

+ xRbF2 1 xRbF1500 J mol

1 . 66

For a description of the excess Gibbs energy of theRb,CsF liquid solution a modified quasichemical modelbased on the quadruplet approximations proposed by Peltonand co-workers39,40 which is well suited for ionic liquids has

been used. The optimization of the excess parameters wasbased on the results of the MD study see Sec. IV whichsuggested slight positive excess enthalpy and little or no ex-cess entropy. We choose to describe the excess Gibbs energyof the Rb,CsF liquid solution by a single, temperature in-dependent parameter, optimized in this study to the value

gRbCs/FF = 188.3 J mol1 . 67

The same notation as proposed by Chartrand andPelton41 is kept here. A comparison between the mixingenthalpy obtained from the MD study and the calculated data

based on our thermodynamic assessment are given in Fig. 6and show an excellent agreement. The gRbCs/FF term whichis the matter of optimization when using the modified qua-sichemical model is in fact equal to the Gibbs energy differ-ence of the exchange reaction of the Rb and Cs atoms on thecation sublattice as shown by

Rb F Rb + Cs F Cs

2Rb F Cs gRbCs/FF. 68

Hence it is related to the short-range ordering of theRb,CsF liquid solution. In other words only whengRbCs/FF=0 is the solution characterized by a random mix-ing and the configurational entropy is equal to the ideal mix-ing term xRbFR ln xRbF xCsFR ln xCsF. Any nonzero valuealways results into ordering of the solution. However, sincethe gRbCs/FF value obtained in our study is very small seeEq. 67 the ordering effect is negligible and the configu-rational entropy is close to ideal. For example, the calculatedmixing entropy at xRbF=0.5 and T=1000 K deviates fromthe ideal behavior only by 0.0003 J mol1 K1.

Furthermore the modified quasichemical model re-quires the definition of the coordination numbers of theRb and Cs atoms in the unary and binary interactions.

They were taken from our previous study3

and are set toZRbRbRb =ZCsCs

Cs = ZRbCsRb =ZRbCs

Cs =6.The thermodynamic data of the solid and liquid phases

TABLE V. Gibbs energy equations of the solid and liquid phases of the pure RbF and CsF compounds.Coefficients are in J mol1. All data are valid for temperature range 2982000 K.

Phase a bT cT ln T dT2 eT1

RbF s 573 639 213.919 42.343 1.30121002 23 450RbF l 563 539 390.427 71 CsF s 569 439 230.509 46.811 8.79501003 4 111CsF l 565 798 405.400 74.057

FIG. 7. Radial distribution functions for the xCsF=0.0, 0.5, and 1.0compositions.

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12/14

of pure RbF were taken from JANAF tables42 as well as thedata for the solid phase of CsF. However, the data of the CsF

liquid phase had to be modified in order to reproduce itsmelting temperature, T=961 K compared to 975 K accord-ing to Ref. 42, measured in this study. The Gibbs energies ofall four phases as a function of temperature GT =a + bT+ cT ln T+ dT2 +eT1 are given in Table V.

The RbFCsF phase diagrams have been calculated byminimization of the Gibbs energy with respect to the propor-tion of the solid and liquid phases. This procedure yields thesolidus and liquidus curves and it is carried out using theFACTSAGE software.43 The resulting phase diagram based onthe BW approximation is shown in Fig. 8 and that based onthe Guggenheim approximation is shown in Fig. 9. Both

phase diagrams are characterized by a minimum on the liqui-dus curve found at similar temperatures and compositions. Inthe case of the BW approximation the minimum is atT=947 K and xRbF=0.274, whereas with the Guggenheimapproximation it is located at T=951 K and xRbF=0.250.However, from a comparison of both figures it is evident thatin the case of the BW approximation a miscibility gap with acritical temperature T=368 K at xRbF=0.575 appears,whereas in the case of the Guggenheim model no immiscibleregion in the solid state has been found. This is due to thefact that in the former case the excess Gibbs energy is as-sumed to be temperature independent and thus has alwaysthe same value for a given composition. Consequently at low

temperatures where the ideal mixing term xRbFRT ln xRbF+ xCsFRT ln xCsF has relatively small influence, the positiveexcess term is dominant and phase separation occurs. This isnot the case with the Guggenheim approximation which doesevolve with temperature. Hence the positive excess param-eters are small at low temperatures while they increase withincreasing temperature and reproduce nearly the same mini-mum on the liquidus curve as found from the assessmentbased on the BW model.

Due to the lack of experimental data at low temperatureit is very difficult to judge which phase diagram is closer toreality; however, since the Rb+ and Cs+ cations are highlyionic and their sizes are very similar it is not expected that a

miscibility gap would appear in the solid state. Thus thephase diagram optimized based on the data of the Guggen-

heim approximation as shown in Fig. 9 is to be preferred.

VI. CONCLUSIONS

The RbFCsF pseudobinary phase diagram we havemeasured using DSC is in very good agreement with the onewe have calculated using our multiscale approach involvingexclusively, for the smaller scale of description, DFT. Thecalculated excess enthalpy reveals an almost ideal behaviorof the liquid phase whereas a small but significant positiveexcess enthalpy occurs in the solid phase. This is a commonfeature in systems with extended solubilities and it illustrates

the similarity of the end members. As for the solid solutionthe calculation reveals that some short-range order is present,quantified by the difference between the zero-order BW andthe first order Guggenheim approximations, and that its ne-glect would lead to the appearance of a miscibility gap in thesolid phase below 367 K in the calculated phase diagram.Beyond this would-be miscibility gap, which is not amenableto experimental confirmation, the short-range order does nothave any significant influence on the melting transitioncurves.

From a more fundamental point of view we have dem-onstrated the applicability of a DFT-based approach to cal-culate the relatively small excess free energies that determine

the phase diagram of a pseudobinary fluoride system. TheSAM proves to be both flexible enough to accommodate thesubtlety of interactions in the solid phase and quite efficientin terms of the amount of input DFT data it requires. Accord-ing to the literature the SAM has been used extensively inthe past to rationalize measured macroscopic thermodynamicdata. We show here that SAM and DFT make up an idealcombination since they are both based on configurationalenergieswhich are not accessible experimentally. As forDFT-based MD with polarizable ions, the technique usedhere had already been validated in several instances and thepresent work shows that it has become a routine and reliabletechnique.

FIG. 8. RbFCsF phase diagram assessed using the BW model solid linesand comparison to the solidus and liquidus data measured in this study byDSC. b Solidus data determined from the heating run; liquidus datadetermined from the cooling curves.

FIG. 9. RbFCsF phase diagram assessed using the Guggenheim modelsolid lines and comparison to the solidus and liquidus data measured inthis study by DSC. b Solidus data determined from the heating run; liquidus data determined from the cooling curves.

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ACKNOWLEDGMENTS

O.B. acknowledges the European Commission for sup-port given in the frame of the program Training and Mobil-ity of Researchers. The ACTINET network for actinide sci-ences, supported by the European Commission in the 6thFramework Programme Contract No. FP6-508836, isthanked for the fellowship provided in order to perform thisstudy JRP 06-020. M.S. acknowledges financial support of

PCR-RSF Programme concert de recherchesRacteur sels fondus.

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Cambridge University Press, Cambridge, 2004.5 J. C. Mathieu, F. Durand, and E. Bonnier, J. Chim. Phys. Phys.-Chim.

Biol. 62, 1289 1965.6 W. L. Bragg and E. J. Williams, Proc. R. Soc. London, Ser. A 145, 699

1934.7 E. A. Guggenheim, Mixtures Clarendon, Oxford, 1952.8 M. Salanne, C. Simon, P. Turq, R. J. Heaton, and P. A. Madden, J. Phys.

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1177 2008.10M. Salanne, C. Simon, P. Turq, and P. A. Madden, J. Fluorine Chem.130, 38 2009.

11R. G. Samuseva and V. E. Plyushchev, Russ. J. Inorg. Chem. 10, 6881965.

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15

H. J. Monkhorst and J. O. Pack, Phys. Rev. B13

, 5188 1976.16 MATERIALS STUDIO v. 4.3, Accelrys Software Inc. http://www.accelrys.com.

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22 J. C. Mathieu, F. Durand, and E. Bonnier, J. Chim. Phys. Phys.-Chim.Biol. 62, 1297 1965.

23 It can be shown that the configurational energies are only sensitive to thesum 1+1. We may thus arbitrarily set 1=0 but we keep it in thealgebraic formulas to preserve the symmetry.

24 The proof in Ref. 5 is a bit difficult to follow but can also be obtainedsimply as follows. Considering a in Eq. 38 as an independent set ofvariables and writing down formally the differential ofF in variables aand T, it is easy to recognize that due to the Guggenheim approximationleading to Eq. 36 the term F/ada in dF includes the logarithm of

the left side of Eq. 35, so that this part of the differential vanishesidentically. Therefore the entropy S = F/T can indeed be computedfrom Eq. 38 as if adid not depend on T.

25 The polynomial denoted S by Mathieu et al. see Ref. 5 is denotedhere by Z to avoid confusion with entropy.

26SCILAB, an open source platform for numerical computation see http://www.scilab.org.

27 As an example, let us consider a configuration with NRb= NCs and M12=NRb. Then on the model side Eqs. 14 and 15 are compatible withP12= NCs, but on the other side reality is not: Some Cs cations, due to thefact that they are near neighbors of an Rb cation fully surrounded by Cs,have at least three Cs near neighborswhich implies P30. This leadsto the contradiction P3+ P12NCs.

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cording to . The explanation of this paradox is that 2 is then slightly

different from zero and in fact both 2 and 2 play a role in short-rangeorder. 2 and 2 would cancel simultaneously only if the solution wassymmetric but this is not quite the case here, and on the other hand ifthere was a large difference between 2 and 2 we would even have tochange the definition of SRO to reliably account for the short-rangeorder.

30 P. A. Madden and M. Wilson, Chem. Soc. Rev. 25, 339 1996.31 K. T. Tang and J. P. Toennies, J. Chem. Phys. 80, 3726 1984.32 A. J. Stone, Theory of Intermolecular Forces Oxford University Press,

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36

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FACTSAGE 5.3, software for thermochemical calculations see http://www.factsage.com.

http://dx.doi.org/10.1016/j.calphad.2007.07.006http://dx.doi.org/10.1016/j.calphad.2007.07.006http://dx.doi.org/10.1098/rspa.1934.0132http://dx.doi.org/10.1021/jp061002uhttp://dx.doi.org/10.1021/jp061002uhttp://dx.doi.org/10.1021/jp075299nhttp://dx.doi.org/10.1524/zkri.220.5.567.65075http://dx.doi.org/10.1103/PhysRevLett.77.3865http://dx.doi.org/10.1103/PhysRevB.13.5188http://dx.doi.org/10.1103/PhysRevB.41.7892http://dx.doi.org/10.1002/(SICI)1097-461X(2000)77:5%3C895::AID-QUA10%3E3.0.CO;2-Chttp://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://dx.doi.org/10.1103/PhysRev.138.A1384http://dx.doi.org/10.1103/PhysRev.138.A1384http://dx.doi.org/10.1039/cs9962500339http://dx.doi.org/10.1063/1.447150http://dx.doi.org/10.1039/b300319chttp://dx.doi.org/10.1021/jp061000+http://dx.doi.org/10.1063/1.467468http://dx.doi.org/10.1063/1.456567http://dx.doi.org/10.1021/jp067073ahttp://dx.doi.org/10.1007/s11661-001-0229-0http://dx.doi.org/10.1007/s11661-001-0228-1http://dx.doi.org/10.1007/s11661-001-0228-1http://dx.doi.org/10.1007/s11661-001-0229-0http://dx.doi.org/10.1021/jp067073ahttp://dx.doi.org/10.1063/1.456567http://dx.doi.org/10.1063/1.467468http://dx.doi.org/10.1021/jp061000+http://dx.doi.org/10.1039/b300319chttp://dx.doi.org/10.1063/1.447150http://dx.doi.org/10.1039/cs9962500339http://dx.doi.org/10.1103/PhysRev.138.A1384http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://dx.doi.org/10.1002/(SICI)1097-461X(2000)77:5%3C895::AID-QUA10%3E3.0.CO;2-Chttp://dx.doi.org/10.1103/PhysRevB.41.7892http://dx.doi.org/10.1103/PhysRevB.13.5188http://dx.doi.org/10.1103/PhysRevLett.77.3865http://dx.doi.org/10.1524/zkri.220.5.567.65075http://dx.doi.org/10.1021/jp075299nhttp://dx.doi.org/10.1021/jp061002uhttp://dx.doi.org/10.1021/jp061002uhttp://dx.doi.org/10.1098/rspa.1934.0132http://dx.doi.org/10.1016/j.calphad.2007.07.006http://dx.doi.org/10.1016/j.calphad.2007.07.006


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