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MODELLING CHEMICAL EQUILIBRIUM PARTITIONING WITHTHE GEMS-PSI CODE

D. Kulik, U. Berner, E. Curti

Sorption, co-precipitation and re-crystallisation are important retention processes for dissolvedcontaminants (radionuclides) migrating through the sub-surface. The retention of elements is usuallymeasured by empirical partition coefficients (Kd), which vary in response to many factors: temperature,solid/liquid ratio, total contaminant loading, water composition, host-mineral composition, etc. The Kdvalues can be predicted for in-situ conditions from thermodynamic modelling of solid solution, aqueoussolution or sorption equilibria, provided that stoichiometry, thermodynamic stability and mixing propertiesof the pure components are known (Example 1). Unknown thermodynamic properties can be retrievedfrom experimental Kd values using inverse modelling techniques (Example 2). An efficient, advanced toolfor performing both tasks is the Gibbs Energy Minimization (GEM) approach, implemented in the user-friendly GEM-Selector (GEMS) program package, which includes the Nagra-PSI chemical thermodynamicdatabase. The package is being further developed at PSI and used extensively in studies relating tonuclear waste disposal.

1 WHAT IS EQUILIBRIUM PARTITIONING?The term Equilibrium Partitioning denotes the(equilibrium) distribution of a chemical element (M)between two phases of variable composition – usuallya mineral solid solution and an aqueous solution. Ameasurable quantity called partition coefficient isdefined as:

[ ][ ]AQ

SM

M=Kd (1)

Here, [M]S is the concentration of M in the solid phase(S), and [M]AQ is that in the aqueous solution (AQ).Except for very simple systems, K d is usually acomplex function of temperature T, solid/liquid ratios/l, total element inventory M TOT, aqueouscomposition (pH, Eh, ionic strength I, concentrationsof complexing ligands [ L ]), and host-mineralcomposition. Measured Kd values may also beinfluenced by sorption or precipitation kinetics. Thus,Kd is a conditional constant, i.e. it applies only tospecific experimental conditions, and cannot begeneralised. All this makes the empirical Kd value notthe perfect choice for long-term predictions of traceelement distributions; hence, more fundamentaltheoretical approaches are necessary.Calculations of aqueous speciation and saturationindices of pure solids, like MCO3 and MOOH, caneasily be performed using widespread computermodels, such as PHREEQC [1], but they are notalways helpful in understanding the relationshipsbetween Kd and system variables. Experimental Kdvalues for trace metals are often significantly differentfrom theoretical partition coefficients obtained fromsolubility products of pure solids. This fact indicatesthat the aqueous concentration of metals is not alwayscontrolled by simple, pure solid equilibria, but ratherby other retention mechanisms involving the hostmineral phases — namely, sorption, re-crystallisationor co-precipitation. In all these cases, the concept of a

fixed thermodynamic solubility fails because ofvariable compositions of both solid and aqueousphases; dissolved [M]AQ would no longer be thesolubility in the classical sense of the word. Attemptsto use variable solubility products depending on thecomposition of aqueous solution, or of a mixed solid,were disappointing, because the law of mass-actionalone does not seem to be sufficient for solving thesolid-solution aqueous-solution (SSAS) equilibria.Some supporting tools, like the Lippmann functionsand diagrams [2,3], can help in binary systems, butnot in higher-order systems [4], or if two or more solidsolutions are involved.Hence, for an adequate thermodynamic description ofpartitioning, it is necessary to go back to the morebasic concept introduced by Gibbs, which states thatthe chemical potential of M is the same in all co-existing phases at equilibrium. The equilibrium state isdetermined by finding mole amounts of all chemicalspecies in all phases such that the total Gibbs freeenergy of the system is minimal at the given statevariables (temperature T, pressure P, bulk molecomposition vector b). In this approach, a variable-composition phase is fully defined by stoichiometryand mole amounts of its end-members (components,species), which need not necessarily exist as puresubstances. The stability of each end-member isgiven by its standard molar Gibbs free energy Go offormation from chemical elements. A deviation fromideal mixing with other end-members (excess partialmolar Gibbs energy, glnRTGGG Ex

idealreal ==- ) isdescribed by the activity coefficient g, a function of theactual phase composition.

2 METHODS OF SPECIATION CALCULATIONSTwo numerical methods of chemical thermodynamicmodelling can be applied to heterogeneous aquaticsystems: (i) Law-of-Mass-Action — Reaction Stoichio-metry (LMA), and (ii) direct Gibbs Energy Minimisation

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(GEM). The LMA approach is common, and availablein many speciation codes, such as PHREEQC [1],MINEQL [5] or EQ3/6 [6], some equipped with databases of thermodynamic constants valid up tohydrothermal conditions. The GEM approach,represented by codes such as ChemSage/FactSage[7,8], GIBBS/Hch [9], Selektor [10,11] or GEMS-PSI,is less common, but is becoming increasingly popular.

2.1 LMAThe LMA approach is extensively described intextbooks [12,13]. Briefly, it requires nothermodynamic data for the “master” species (usuallyaqueous ions), but only the logK of formation of the“product” species at the P, T of interest.

To find the “speciation”, i.e. equilibrium concentrationsof aqueous ions or complexes, surface species andsolids included, LMA codes usually employ theNewton-Raphson method. This iteratively solves asystem of material balance equations, with non-linearboundary conditions being in the form of mass actionequations for product species [1]. The LMA algorithmactually minimises the material balance residuals tovery good precision: 10-18 ÷ 10-21 molal, relative to themaster species.Unfortunately, the LMA algorithm in its common formhas some serious limitations in setting up and solvingmulti-phase models involving partitioning. Namely: (i)only one variable composition phase is tolerated in themass balance (usually the aqueous electrolyte) — allother phases (pure solids, solid solutions) must betaken at fixed compositions, and with constantsolubility products; (ii) usually, stable solids must beknown in advance, in order to be included into themass balance; (iii) surface complexes are treated inthe same way as aqueous complexes, but are subjectto additional balance constraints on total amount ofsurface sites and, optionally, to electrostatic correctionterms; and (iv) redox couples must be set at input byassigning different redox states of the same element,either by separate mass-balance constraints (e.g. forFeII and FeIII), or by the input pe or Eh of the aqueoussolution.

2.2 GEMThe GEM approach is based on a mass balance forthe entire system, which is set up by specifying thetotal amounts of chemical elements and a chargebalance only. These elements and electric charge arecalled “Independent Components” (ICs, belonging tothe set N). All chemical species, in all phases, arecalled “Dependent Components” (DCs, belonging tothe set L), since their stoichiometries can be built fromICs. To some extent, ICs can be compared with LMA“master species”, and DCs with the “product species”.A significant difference between them lies in theexplicit definition of the thermodynamic phases(belonging to the set F, each including one or more

DCs and additional properties, such as the specificsurface area) in the GEM system formulation. EachDC is provided at input with its elementalstoichiometry, and a value of the standard molar (orpartial molal) Gibbs energy Go, which is taken fromthe database and corrected to the P, T of interest, ifnecessary.In the GEM method, the activities and concentrationsof the DCs are treated separately for each phase,taking into account appropriate standard/referencestates and activity coefficients. The equilibriumassemblage conforming to the Gibbs phase rule canbe (in principle) selected automatically from a large listof stoichiometrically possible phases. The equilibriumpartitioning in a multiphase system, including forexample an aqueous solution, a gas mixture, one orseveral solid solutions, and, optionally, sorptionphases, is computed simultaneously for all phases ina straightforward way. The GEM “Interior PointsMethod” (IPM) algorithm [10,11] does all this because,in addition to the speciation vector x (mole amounts ofDCs – the primal solution), it computes simultaneouslya complementary dual solution vector u (holdingequilibrium chemical potentials of ICs at the state ofinterest). As shown below, the power of GEM IPM liesin comparing the DC chemical potentials obtainedfrom primal x and dual u vectors, wherever possible.

3 GEM CONVEX PROGRAMMING METHOD3.1 Mathematical FormulationThe goal of GEM is to find a vector of the DC moleamounts, x ={xj, jΠL}, such that:

G(x) fi min, subject to Ax = b (2)

where A = {aij, i ΠN, j ΠL} is a matrix of formulastoichiometry coefficients of the i-th IC in the j-th DC,b = {bi, i ΠN} is the input vector of total moleamounts of IC, and G(x) is the total Gibbs energyfunction of the whole system:

Fu ŒŒ=  kLjxxG kk j

jj ,,)( (3)

In Eqn. (3), Lk is a subset of DC in the k-th phase, anduj stands for the dimensionless chemical potential ofthe j-th DC:

kFjj

j,Tj LjconstCC

RT

GŒ++++= ,lnln

o

gu (4)

where oTjG , is the standard molar Gibbs energy,

R!=!8.3145 J⋅K-1⋅mol-1 is the universal gas constant, Tis temperature (K), Cj = f(xj) is concentration, jg isthe activity coefficient of the j-th DC in its respectivephase, CF stands for the Coulombic term (for chargedsurface complexes), and const converts from thepractical to the rational (mole fraction) standard-state

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concentration scale. The more detailed theory, withexpressions for uj in aqueous, gaseous, solid/liquidsolution and sorption phases, can be found in[10,11,14].The IPM is a non-linear minimisation algorithmspecifically developed to solve chemical equilibriumproblems involving many potential single- and multi-component phases. This “engine” of the Selektormodelling codes [15] finds simultaneously two vectors,the primal x and the dual u optimal solutions of theproblem (2), by checking the Karpov-Kuhn-Tucker(KKT) necessary and sufficient conditions for G(x) tobe minimum [10]. The KKT conditions can be writtenin the integrated vector-matrix form for the case ofcomplete equilibrium as:

( ) 0ˆˆ

;0ˆ;ˆ

;0ˆ

=-

≥=

≥-

uAõx

xbxA

uAõ

TT

T

(5)

where the superscript T represents the transposeoperator. When rewritten with indices for a j-thspecies:

 Œ≥-i

iijj Niua ,0u (6)

the first KKT condition (Eqn. 5) implies that, for any j-th species (DC) present at equilibrium concentrationCj in its phase, the primal chemical potential uj

(calculated from the equilibrium mole amount jx and

the standard molar Gibbs energy Gjo according to

Eqn. 4) is equal to a dual (stoichiometric) chemicalpotential

 Œ=i

iijj Niua ,h (7)

Hence, dual solution values uj are chemical potentialsof independent components (IC) at the equilibriumstate of interest, i.e. at given P, T and b of the system.

The last “orthogonality” KKT condition in Eqn. 5 helpsto zero off molar amounts jx of unstable species forwhich the first KKT condition (Eqn. 6) cannot be met,because of either the mass-balance (secondcondition) or non-negativity (third condition)constraints. It is, in fact, this last condition that givesrise to the name “Selektor”, because the orthogonalityconstraint enables the algorithm to “switch off”unstable species and phases.The KKT conditions were also extended for the casefor which the sought-after mole amounts jx of someDCs are subject to the metastability constraints [11].This makes it possible to use the GEM IPM algorithmfor simulating various kinetically-dependentprocesses, such as sequences of partial equilibriumstates.

Overall, the KKT conditions, and the related vector-matrix notation, can be regarded as a very condensedand precise representation of equilibria in isobaric-

isothermal, heterogeneous, multiphase systems ofany complexity and size. A large number of multi-component phases can be included simultaneously inthe initial approximation. Such equilibria can be foundusing the IPM algorithm, as long as the internally-consistent, standard-state molar properties of the DCsat the P,T of interest, and the equations for thecalculation of the activity coefficients in the phases,are all provided. In many cases, the inconsistent inputstoichiometries and thermodynamic data can bedetected automatically.

3.2 Dual ThermodynamicsFor any species, in any phase present at equilibrium,Eqns. (4,6) can be combined into a DualTh (“dualthermodynamic”) equation that compares “dual” and“primal” DC chemical potentials as

jj uh = , or

 ++++=i

Fjjj,T

iij CCconstRT

Gua glnln

o

(8)

The difference RT

GoTj

j,-h defines the activity of the

j-th DC. Hence, the activity of any chemical speciescan be retrieved from its o

TjG , value, the chemical

formula, and the dual solution vector u . Indeed,Eqn.!(8) is used within the GEM-Selektor code tocompute: (i) activities of gases, aqueous, solid-solution and surface species; (ii) saturation indices forsingle-component condensed phases; and (iii) activityfunctions such as pH, pe and Eh. For instance, inaquatic systems, Eh (in Volts) is computed [11]according to:

ee uTF

RTu ⋅⋅== 000086.0Eh (9)

where ue is the dual chemical potential relating to thecharge balance constraint, and F is Faraday’sconstant. Likewise, pH is computed without using themolality or the activity coefficient of the H+ ion:

pH = -0.4343⋅(uH + ue) (10)

in which the coefficient -0.4343 derives from theconversion ln to inverse l o g10 scale. Hence, theequilibrium values of redox potential and pH can befound in a simple and general way, without applyingmore specific and complex electrochemicalrelationships.Another application of Eqn. (8) is gaining moreimportance in “inverse modelling”, which is aimed atretrieving of thermodynamic data from theexperimental equilibrium partitioning data. The idea[11] of such calculations, which can be termed “dual-thermodynamic” (DualTh), is quite simple. Supposethat bulk compositions, for example of an aqueouselectrolyte, and the equilibrated solid solution phasehave been measured. Then, the u vector of dual

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potentials ui can be computed using GEM IPM bymodelling equilibrium speciation in the aqueousphase, but without considering the solid solutionphase of interest in the mass balance. Next, ui valuescan be used along with the concentration C j

(calculated from the measured bulk composition ofsolid solution phase and the j-th end-memberstoichiometry) to find one unknown parameter on theright-hand side of Eqn. (8).

Fig. 1: Screen image of GEMS dialogue for selecting independent components to create a new modelling project.

If the oTjG ,

value is known, the activity coefficient jg of

the j-th end-member can be found, and convertedinto parameter(s) of the chosen mixing model [4].Otherwise, the unknown o

TjG , value can be

determined for a specific mixing model [16]; see alsoExample 2 below. If several experimental points atdifferent compositions are available, then anunlimited number of alternative end-memberstoichiometries can be compared, leading toidentification of an “optimal” stoichiometry, including a

oTjG ,

estimate and its uncertainty. This technique canalso be helpful in the planning of new co-precipitationexperiments, and, at a later stage, in interpreting theirresults.

4 GEM-SELEKTOR V.2-PSI PACKAGEGEM-Selektor (GEMS) is a user-friendly geochemicalmodelling package in which the GEM method isimplemented. In particular, GEMS-PSI inherits theIPM non-linear minimisation algorithm from Selektorcodes developed since 1973 in Russia, and since1991 internationally.

The GEMS package, written in C/C++, combines thehigh-precision IPM-2 module [15] with software toolsfor setting up and running the user’s modellingprojects, managing built-in thermodynamicdatabases, and re-calculating thermodynamic data.All data and controls are accessible through amodern graphical user interface (GUI), based on thecross-platform Qt® toolkit [17], thanks to which thecode compiles on Windows®, Linux®, Mac OS X,and other platforms. The GEMS-PSI package (atpresent, Windows, and soon Linux, versions) can bedownloaded free of charge from the following webpage: http://les.web.psi.ch/Software/GEMS-PSI. Thepackage has been made available to a broadresearch community in the hope of collecting userfeedback, and thereby to continuously improve thecode and documentation.The setup of new modelling projects, phases andspecies is facilitated by the GEMS GUI (Fig. 1), whichalso includes a run-time help browser. The inputstandard-state thermodynamic data are automaticallycorrected to the temperatures and pressures ofinterest using well-established techniques

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appropriate for solids, gases, aqueous and surfacespecies. In addition, GEMS can also simulate variousirreversible mass-transfer processes, such astitrations, mixing, weathering or sequential reactors,from the principles of local and partial equilibrium.Results of such "process simulations" are stored inthe project data-base, and can easily be tabulated orplotted at run-time (Fig. 2), or exported to text files.

4.1 GEM-Selektor IPM-2 ModuleThe sensitivity (minimum mole amount of IC) andprecision (maximum mass-balance residual) of theoriginal IPM algorithm [10] were not always sufficientfor modelling systems with trace radionuclideconcentrations. Hence, a collaboration project wasundertaken in 2001-2002, which resulted in animproved IPM-2 module [15]. The sensitivity andprecision parameters achieved in IPM-2 are usuallyas good as those in LMA codes. The precision thatcan actually be attained depends on the internalconsistency of the input thermodynamic data, and tosome extent also on the bulk composition (bufferingcapacity) of the chemical system. The IPM-2 modulealso converges well with highly non-ideal systems,including solid solutions, or electrostatic surfacecomplexation on heterogeneous mineral-waterinterfaces.

4.2 Thermodynamic DatabaseThe GEMS package includes two built-in chemicalthermodynamic databases, compiled from Nagra/PSI01/01 [18] and SUPCRT92-98 [19] data sets. Thenecessary data is automatically selected, and thencopied into the modelling project database, when theuser sets up a new modelling project (see Fig. 1).There, the data will become immediately available forthe calculation of the equilibrium states, afterproviding a “recipe”: i.e. the bulk composition of thesystem, optionally including the non-ideality andmetastability parameters.!Later, the user can easilyextend the project database with new dependentcomponents and phases, as well as replace somerecords with his/her own input thermodynamic data, ifneeded and justified. Only the Nagra/PSI 01/01database is officially supported by PSI/LES [20];thermodynamic data from other sources can beinvolved at the user’s discretion only.To extend the range of applications of GEMS-PSIbeyond low-temperature aquatic systems relevant tonuclear waste disposal, a “third-party database”collaboration strategy has been developed. In such acollaboration, an external research team maintains aweb page, with a specific thermodynamic database,in GEMS format. Anyone can download it, and thenuse it as a “plug-in” in the already installed GEMS-PSI package, or instead of the original built-indatabase. In this case, the only task required of theGEMS-PSI Development Team is to make sure thatthe third-party database has been correctly imported

in the GEMS format, and that all calculationspertinent to that database are correctly performed bythe GEMS code.

4.3 Graphical User Interface (GUI)The user-friendly GUI is essential in improving thequality and acceptability of the interpretation of themodelling results, and in saving the researcher’sworking time. The GEMS GUI, together withdatabase management modules and screen forms,tool tips, runtime help browser and onlinedocumentation, makes the set-up or modification ofmodelling problems quick and easy. Almost allmodelling projects eventually need to be extended toinclude new phases and chemical species,sometimes during the course of the project. The dataaccess screen forms for chemical species andphases, with the validation subroutines behind them,facilitate much of this tedious work.A simple graphical presentation module in GEMS isvery helpful in sampling selected modelling results,and presenting them as multiple plots (Fig. 2). Theuser defines what to sample, and how to re-calculate,by writing basic “math scripts”, which areautomatically translated and executed. The sampleddata can be exported as ASCII files of any formatwith the help of “printing scripts”. An on-linescreenshot tutorial helps users learn how to operateGEMS for real modelling examples in a few hours.GEMS stores any modelling exercise in a separate“modelling project” directory, which can be zippedand shared with other users, or sent to thedevelopment team if something went wrong.

Fig. 2: Screen image of graphical output from GEMS“process simulation” of Sr isotherm in bariteat P=1 bar, T=25°C. The ordinate “log_m” isthe logarithm of the total dissolved molality[M], and the abscissa “X_cel” is the logarithmof the mole fraction of the celestite (SrSO4)end-member in a (Ba,Sr)SO4 regular binarysolid solution with the Margules parametera=!2.

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Of course, user-friendliness alone cannot replacescientific efficiency in the GEM method, a point whichis illustrated below in terms of two examples from ourown usage of the code. Example 1 illustrates theprediction capabilities of solid-solution aqueous-solution models in applications relevant to nuclearwaste disposal. Example 2 demonstrates howGEMS-PSI and DualTh calculations can be used inthe interpretation of three independent experimentalpartitioning data sets. This leads to a newthermodynamic solid-solution aqueous-solutionmodel of europium incorporation in calcite, applicablefor a wide range of compositions at ambienttemperatures. Another example of the power ofGEM-Selektor in modelling a complex, redox-sensitive, solid-solution aqueous-solution hydro-thermal system can be found in a recent PSI Report[21]. More applications of GEMS are currently inprogress.

5 EXAMPLE 1: PREDICTING IMPROVEDRETENTION OF RADIUM IN A NUCLEARWASTE REPOSITORY

5.1 Background and AimsSafety analysis studies for the SwissHigh/Intermediate Level Waste (HLW/ILW) repositoryhave clearly demonstrated that, in specific scenarios,the expected maximum concentration of 226Ra mightbecome a critical issue. This is particularly true whenthe “classical” approach, which relies on solubilitycontrol by pure solid phases, is used to evaluatemaximum dissolved nuclide concentrations. Theexample here presents this “classical” approach inthe context of the repository chemical systemsketched in Fig. 3.Dissolved radium concentrations may be stronglyaffected by co-precipitation with barium and strontiumsulphate minerals, c.f. [22], which are present asimpurities in the repository backfill and host rock.Based on the more detailed inventory analysis, asimple solid-solution-based model accounting for thiseffect has been proposed and evaluated using GEMS[23]. In this way, the beneficial impact of the bariumsulphate inventory on dissolved radiumconcentrations has been demonstrated.

Fig. 3: Schematic of the repository (chemical)system addressed in Example 1. Propertiesof the pore water relevant to the study aregiven in the lower left corner.

5.2 ”Classical” Solubility Calculation for RaThe “classical” approach assumes that the totaldissolved concentration of radium [Ra]AQ is definedby an equilibrium with the least soluble pure Ra solid.A simple operational method has been adopted tocalculate this solubility, starting from a pore solutionin equilibrium with compacted bentonite, the backfillmaterial for the planned Swiss repository (briefindications are given in Fig. 3, and more details areprovided in [24]). In our model, an increasing amountof RaCl2 "spike" is “added” to the pore-water/bentonite system. The radium solubility limit isattained when, according to the contents of thethermodynamic database, saturation with the moststable (i.e. least soluble) Ra solid has beenestablished. Since the reference solution itself doesnot change to any significant degree, this procedureis the method of choice when trace concentrationsare involved. In our specific bentonite pore water, thesolid RaSO4 starts precipitating when the “added”[RaCl2]AQ reaches 4.8·10–8 M (mol⋅L-1). Thus, thesolubility limit of Ra in the reference bentonite porewater is just 4.8·10–8 M.The distribution of Ra species in solution (i.e. thespeciation) is dominated by the mono-sulphatecomplex and the free aqueous ion, plus a few percentof the mono-chloride complex:

RaSO4(aq): 3.1·10–8 MRa2+: 1.6·10–8 MRaCl+: 4.8·10–10 M

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This calculated maximum concentration is inverselyproportional to the sulphate ion concentration (about60 mM), and may thus be affected by all the variableswhich influence [SO4

-2]AQ: e.g. basic system definition,gypsum formation, sulphate reduction, etc. Up to thispoint, the conventional LMA geochemical speciationcodes (Sec. 2.1) are sufficient (and well suited) toperform all calculations. The GEMS code performssimilarly.

5.3 Model Concepts and Material BalancesIn order to establish a solid solution model forpredicting the behaviour of radium, it is essential toevaluate the detailed material balances in thesystem. Each waste canister, which contains thereprocessed waste of 1.8 tons of UO2 spent fuel,produces a maximum inventory of 0.0128 moles of226Ra within 300!000 years after discharge from thereactor (cf. [25]). According to present layouts, eachcanister is surrounded on average by 57.4 tons of(dry) bentonite and 12.7 tons of pore solution.The basic ideas for establishing a solid solutionmodel from these facts are rather simple.

• The chemical behaviour of Ra is stronglyrelated to that of Ba and Sr, due to well-established similarities in the chemicalproperties of these elements.

• The bentonite surrounding the wastecontains minor impurities of Ba (and Sr)sulphates. However, compared to Ra, theseimpurities represent large quantities,considering there is 57.4 tons of bentonite.Further details may be taken from [23].

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• In consequence of a very slow fluid motionthrough compacted bentonite, the chemicalsystem has sufficient time to re-equilibrate (ofthe order of 105 years).

The radium mobilised from the waste (canister) isexpected to interact with the barium/strontiumsulphates (barite/celestite), present in the bentoniteas impurities. Thus, radium is assumed to form asolid solution via re-crystallisation of the sulphates.This view is corroborated by the fact that pore watersof Opalinus Clay formations (the present host rockcandidate for a Swiss repository) are saturated withrespect to barite and celestite.

Table 1: Bulk composition recipe (in moles, water inkilograms) of a simplified chemical system.The resulting equilibrium solution is a Na-Cl-SO4–type water of pH 7.3. Its saltconcentration corresponds to about half ofsea-water salinity.

RaSO4 0.0128BaSO4 71.86SrSO4 38.22Water 12740 kgNaCl 2138.5

CaCO3 8025CaSO4 796.8

Na2SO4 672.4H2SO4 254.8

Air1) large excess1A large amount of artificial “dry air”, including 0.63 mol% ofCO2(g), 80 mol% of N2(g) and 19.4 mol% of O2(g), giving a CO2partial pressure of log10pCO2 =_–_2.2.

Detailed inventories have been derived from traceelement analyses of bentonites, and from thebentonite pore water composition [23,24]. A chemicalsystem normalised to one waste canister has beenconstructed, as outlined in Table 1.

5.4 Results of GEMS Solid Solution ModellingIn order to obtain a point of reference, the chemicalsystem from Table 1 was first examined assumingthat no solid solution is formed at all. As expected,the GEMS calculation yielded the same results asthose previously obtained with the LMA code. Acloser look at the earth-alkali-sulphate system shownin Table 2 reveals that all four sulphates co-exist atequilibrium, and that [Ra]AQ = 4.8·10–8 mol⋅(kg!H2O)-1,as before. Following the outline given above, in thenext step it was assumed that BaSO4(s) andRaSO4(s) form an ideal solid solution. Table 3demonstrates the dramatic impact of solid-solutionpartitioning on the dissolved Ra concentration.Since RaSO4(s) and BaSO4(s) have nearly identicalstructures, and BaSO4(s) is present to large excess(i.e. 5641:1), the ideal mixing reduces dissolved Ra

by about the same ratio: from 4.8·10–8 to 8.6·10–12

molal. The solubility of BaSO4(s), the “excesspartner”, is not changed to any significant degree.Furthermore, using the capabilities of GEMS, wecould easily evaluate the consequences of an idealternary solid solution, including SrSO4(s) as anadditional end-member. In this case, we wouldpredict [Ra]AQ to decrease to 5.6·10–12 molal.

Table 2: GEMS calculation for the simplifiedchemical system given in Table 1, forconvenience scaled down to 1 kg of water.

Amounts of solid at equilibrium Solution[mmol] [mmol⋅(kg H2O)-1]1)

RaSO4 0.956 pure phase [Ra]AQ: 0.0483BaSO4 5640 pure phase [Ba]AQ: 0.0889SrSO4 2886 pure phase [Sr]AQ: 114

CaSO4 70012 pure phase [Ca]AQ: 13’826[SO4

2–]AQ: 65’4321A 0.3!% molality!/!molarity (mol⋅(kg!of!H2O)-1 / mol⋅L-1) differencewas neglected.

It is known that miscibility in the (Sr,Ba)SO4 solidsolution series is rather limited [3], due to the differentionic radii of Sr2+ and Ba2+. As the ionic radii of Ba2+

and Ra2+ are similar, the series (Sr,Ra)SO4 shouldalso have a large miscibility gap at room temperature.Hence, a model with two co-existing, binary solidsolutions!—!ideal (Ba,Ra)SO4, non-ideal (Sr,Ra)SO4— appears to be more realistic than that with theideal ternary solid solution (Ba,Sr,Ra)SO4.

Table!3: GEMS calculation for the simplifiedchemical given in Table 1, assumingBaSO4(s) and RaSO4(s) form an ideal solidsolution.

Amounts of solid at equilibrium Solution[mmol] [mmol⋅(kg H2O)-1]

RaSO4 1.0 end-member [Ra]AQ: 0.0000086BaSO4 5640 end-member [Ba]AQ: 0.0889SrSO4 2886 pure phase [Sr]AQ: 114

CaSO4 70!016 pure phase [Ca]AQ: 13!823[SO4

2–]AQ: 65!417

Some output from a single GEMS calculation, for asystem with both (Ba,Ra)SO4 and (Sr,Ra)SO4 solidsolutions, is shown in Figs. 4 and 5. The resulting[Ra]AQ~!8.2·10–12 molal is almost the same as thatgiven in Table 3, where only the (Ba,Ra)SO4 phasewas involved. Therefore, radium is stronglypartitioned towards barium sulphate, and thepresence of the more soluble celestite plays noimportant role. Radium is not expected to partitioninto the more abundant, but more soluble, gypsum(CaSO4⋅2H2O; 70016 mmol⋅(kg H2O)-1 present). In thebentonite environment, the excess gypsum maintainsa rather high total dissolved concentration of sulphate([ -2

4SO ]AQ~0.066 molal, Fig. 4), which suppresses

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those of [Ba]AQ, [Sr]AQ and [Ra]AQ. Gypsum may becompletely leached out of the system on a relativelyshort time-scale; in this scenario, a correspondingdrop in [S]AQ to 0.004 molal would increase [Ra]AQ by5-7 times only.

Fig. 4: A GEMS screen image, with results for ICs.Column “b” is the input bulk composition ofthe system (moles); “Cb” are the mass-balance residuals (moles); “u” are the dualsolution potentials uj (dimensionless); “m_t”are total dissolved molalities of the ICs, andthe “lgm_t” are their decimal logarithms.

5.5 DiscussionFigure 5 shows in detail how radium is partitionedamong the severa l so lu t ion phases.

In this calculation, the aqueous electrolyte, the gasmixture, the 3 solid solutions, and the 9 single-component solid phases, were all initially included.At equilibrium, 6 phases were all found to be stable,including (Ba,Ra)SO4 and (Sr,Ra)SO4 solid solutions,calcite and gypsum. The pure Ra-sulphate phase is3.77 orders of magnitude undersaturated. The(Ba,Ra)CO3 phase is quite unstable in the presenceof the sulphates.

Figure 6 outlines the impact of solid solutionformation on trace element solubilities. The strongdecrease in the trace element concentration isprimarily an effect of end-member mole fractions,once the solid solution phase is stable. Theproperties of mixing and “common-anion” effects playa rather minor role at trace concentrations, asexemplified by the uncertainty band in Fig. 6Unfortunately, ideal solid solution formation is by nomeans the rule; usually, non-ideal solid solutions,with narrow regions of mixing and miscibility gaps,are formed, but better knowledge of theirthermodynamic properties is largely absent.However, at trace mole fractions, the non-ideality issufficiently well approximated by assuming idealbehaviour, and by taking the stability of the trace end-member as Ex

TroTrTr GGG +=* .

118

Fig. 5: Screen image of GEMS “System” dialogue showing the calculated equilibrium state. Note that more than95% of radium is partitioned into the barite (“x (moles)” column). Activities of RaSO4 species (“log(activity)”column) are equal in both solid solutions containing it, but the concentrations are not equal because of thedifferent stabilities of the major end-members, and the different activity coefficients.

Fig. 6: The impact of (Ba,Ra)SO4 solid solution ondissolved [Ra]AQ as a function of available Bainventory (thick solid curve). Thin solid curvesreflect the solid solution stability uncertaintiesof one order of magnitude. The horizontal bardescribes a “classical” situation, in which[Ra]AQ is controlled by a pure RaSO4 phase.

6 EXAMPLE 2: EUROPIUM INCORPORATIONIN CALCITE

6.1 BackgroundTrivalent europium, EuIII, is a lanthanide elementfrequently used in laboratory studies as an analogueto AmIII, CmIII and PuIII, three safety-relevant actinidesoccurring in various types of radioactive waste. In along-term repository, these actinides will be leachedfrom the waste matrix in trace concentrations, andwill interact with the surrounding materials (e.g.cement and bentonite backfill). The main secondaryproduct of cement alteration is calcite (CaCO3), acommon reactive mineral hosting a variety of traceelements through solid solution formation [26].Calcite is also present as a minor phase in OpalinusClay, the host rock for the planned HLW/ILWrepository in Switzerland. The interaction of traceactinides with calcite rapidly produces dilute solidsolutions, either through co-precipitation from anoversaturated solution, or through re-crystallisation ofpre-existing calcite.Many studies of calcite solid solutions have beenpublished. Yet, the great majority are empirical (Kddeterminations), and focus on the incorporation ofdivalent ions (e.g. Cd2+, Mg2+, Sr2+, Ba2+). Becausethese ions substitute for the equally charged Ca2+ (ahomovalent substitution), the choice of end-memberstoichiometry and structure is rather straightforward.Usually, such divalent metals form pure carbonates,with the same stoichiometry and structure as calcite

(CdCO3, MgCO3, etc.). Hence, the correspondingsolid solution end-members can be easily defined.Because the solubility products, Ko

s, of these purecarbonates have been measured rather accurately,the standard free energy of formation of any divalentmetal carbonate end-member, Go

f (em), can be easilycalculated through the relations00lnRsGRTKD=-

(11)0000()()()RpfrffGGpGrGemnnD=--ÂÂ

(12)

where DGoR is the standard molar free energy of the

end-member dissolution reaction, Gof (r) and Go

f (p)are the free energies of formation of auxiliary reactantand product species, the np,r are stoichiometrycoefficients, and Go

f (em) is the free energy offormation of the solid solution end-member to bedetermined. Once Go

f (em) is known, binary solidsolution models can be developed and comparedagainst experimentally determined trace-metalsolubilities.The situation is far more complex for substituting ionssuch as Eu3+, which have a different charge to that ofCa2+ (a heterovalent substitution). In this case, noobvious end-member stoichiometry can be defined,since formulae and structures of pure Eu-solids donot have much in common with calcite. Moreover,substitution of a trivalent cation for Ca2+ induces alocal charge imbalance in the calcite structure, whichmust be compensated. The situation is made morecomplicated by the fact that different chargecompensation mechanisms exist: for instance, acoupled substitution (Na+ + Eu3+ for 2 Ca2+), or asubstitution of electro-neutral complexes (Eu(OH)3 forCaCO3).In this example, we show how the GEM method hasbeen applied to develop an appropriate solid solutionmodel, capable of explaining a large variety ofexperimental data on co-precipitated or recrystallisedEu calcites, obtained under widely different pH andpCO2 conditions [27]. In order to test the model, weconsidered three sets of data: (1) co-precipitationexperiments at pH ~ 6 and pCO2 = 1 bar, obtained inNa-Ca-HCO3-ClO4 solutions of about 0.1!Melectrolyte concentration [28]; (2) co-precipitationtests in synthetic seawater at pH ~ 8 and pCO2 =0.0003 ÷ 0.3 bar [29]; and (3) re-crystallisation testsin cement pore water at pH ~ 13 and very low pCO2[30].

6.2 Forward Binary ModelsAt the first modelling step, we applied a method verysimilar to that used for divalent cations. Four Eu

Ba/Ra-ratio in solid solution

Dis

solv

ed R

a [m

ol/k

g]

Ba/

Ra

ratio

in s

afet

y re

leva

nt s

yste

m

10-7

100'000

10-8

10-9

10-10

10'0001'0001001010.1

10-11

10-12

Solubility of pure RaSO 4

119

solids exist, with well-known solubility products, fromwhich the standard molar Gibbs free energy offormation for the corresponding Eu end-members canbe derived.As evident from Table 4, the free energy of the majorend-member, Can(CO3)n, must be scaled to thestoichiometric factor n = 1, 2 or 3. Each of the listedend-member pairs can be connected to a specificsubstitution mechanism. For instance, theEuNa(CO3)2—Ca2(CO3)2 solid solution implies acoupled substitution Na+ + Eu3+ = 2 Ca2+, while theEu2(CO3)3 —Ca3(CO3)3 model implies that two of thethree adjacent Ca-sites in calcite are substituted, butthat the third one remains vacant.Table 4: End-members and Gibbs free energies of

formation (kJ⋅mol-1) for the “forward” binaryEu-calcite solid solution models.

Europium side Gof Calcium

sideGo

f

Eu2(CO3)3 -2932.7 Ca3(CO3)3 -3387.6

EuOHCO3 -1383.6 CaCO3 -1129.2

EuNa(CO3)2 -2009.2 Ca2(CO3)2 -2258.4

Eu(OH)3 -1201.0 CaCO3 -1129.2

For the end-member pairs listed in Table 4, four idealbinary solid solution models were developed for theconditions relevant to the selected data. This meansthat four series of model calculations were performedfor each of the three data sets. In each series, theoverall equilibrium between the aqueous solution andthe binary calcite solid solution was calculated fordifferent total amounts of Eu in the system, yieldingEu equilibrium concentrations in the aqueous solutionas functions of Eu mole fraction in calcite (isotherms).For instance, Fig. 7 shows isotherms resulting for thepH~6 data.

Eu (CO )

23 3

Eu OH CO

3

Eu Na (CO )

23

Eu (OH)

3

GEMS model calculationscoprecipitation data, pH 6

mE

u

Euc

10-11

10-10

10-9

10-8

10-7

10-210-310-410-510-610-710-9 10-8

10-6

pure Eu (CO ) stable2 3 3

Fig. 7: Isotherms (lines) for binary Eu-calcite idealsolid solutions (Table 4), compared againstthe co-precipitation data [28] obtained at

pH~6. The symbol cEu is the Eu mole fractionin calcite, and mEu is the equilibrium molalityin aqueous solution.

These results demonstrate that binary solid solutionsinvolving Eu2(CO3)3 or Eu(OH)3 end-members mustbe rejected. The Eu2(CO3)3 end-member must beexcluded because the slope defined by the modelisotherm (+1/2) is not compatible with the trenddefined by the experimental data (slope!+1). Theisotherm slope is related to the stoichiometry of Eu inthe substituted complex. The +1 slope defined by thedata implies substitution of isolated Eu cations incalcite, and excludes substitution of dimeric Eugroups (two Eu ions in adjacent Ca sites), in whichcase, a slope of +1/2 would arise in the isotherm plot.The Eu(OH)3 end-member must also be ruled outbecause of the too large discrepancy between modeland data (non-ideality corrections can account for ashift of, at most, one order of magnitude). Finally, wealso had to discard the EuNa(CO3)2 end-member,because the experimental data do not show theexpected sensit ivi ty to dissolved [Na]AQconcentrations. In conclusion, of the four binarymodels considered, only the EuOHCO3–CaCO3 solidsolution is compatible with the data within a realisticnon-ideal correction.The same four binary models yielded surprisinglydifferent results when applied to the other twoexperimental datasets [29,30] (not shown). It turnedout that only a model involving Eu(OH)3 wouldrealistically explain the data obtained at pH~13,whereas the data obtained at pH~ 8 could beexplained by assuming either Eu(OH)3 or Eu2(CO3)3as end-members.

6.3 DualTh Calculations and a Ternary ModelAs a next step, we decided to test binary solidsolutions with two additional hypothetical end-members, EuH(CO3)2 and EuO(CO3)0.5, for which nopure solids (hence, no Ko

s or Gof values) are known.

We had to resort to an inverse modelling procedure,through which the standard Gibbs energy offormation Go

f of a selected end-member is deriveddirectly from the equilibrium partitioning data. Fromthe results of the GEMS DualTh calculations (seeSection 3.2), applied first to the equilibrated aqueoussolutions, the chemical potentials of the proposedend-members were calculated at each singleexperimental point. With the help of the measured Eumole fractions, free Gibbs energies of formation werethen estimated. A given end-member was consideredto be appropriate if almost the same Go

f va lueresulted for all points of the experimental isotherm. Inpractice, a tolerance of ±2!kJ⋅!mol-1 (±0.35 log Ko

s

units) was set for the standard deviation of the Gof

estimates. The results of such calculations are

120

presented in Table 5, for all six end-membersconsidered so far.

Table 5: Mean end-member standard free energiesof formation G o

f (kJ⋅mol-1) derived fromDualTh calculations for the threeconsidered datasets.

pH~6 pH~8 pH~13

Eu2(CO3)3 -2939.9 ± 4.6 -2921.8 ± 1.4 -3106.6 ± 6.0

EuOHCO3 -1379.0 ± 2.0 -1364.1 ± 1.0 -1425.5 ±1.7

EuNa(CO3)2 -2010.2 ± 2.3 -1994.5 ± 1.0 -2088.8 ± 1.7

Eu(OH)3 -1221.7 ± 2.1 -1192.5 ± 1.0 -1193.3 ± 1.7

EuH(CO3)2 -1775.1 ± 2.0 -1774.6 ± 1.0 -1896.6 ± 1.7

EuO(CO3)0.5 -1063.2 ± 2.3 -1041.0 ± 1.0 -1072.2 ± 1.7

Equivalent Gof values (shown in boldface in Table 5)

result from two out of the three data sets, if eitherEu(OH)3 or EuH(CO3)2 is assumed as a minor end-member of the binary solid solution. This means thatall three sets of experimental data cannot bedescribed simultaneously by any binary model.However, assuming a te rnary solid solutionEuH(CO3)2-Eu(OH)3-CaCO3, with free energies of Euend-members as specified in boldface in Table 5, aconsistent model for all three experimental data setscan be constructed. We have tested this hypothesis,and obtained the results plotted in Fig. 8.Indeed, all three datasets could be reproducedsimultaneously on the basis of the above-mentionedideal ternary solid solution model. The content of Euin the solid solution phase is strongly dominated bythe EuH(CO3)2 end-member in the pH~6 calcites, andby the Eu(OH)3 end-member in the pH~13 calcites.Both Eu end-members are present in comparablemole fractions in the pH~8 calcites. Moreover, wewere able to show that ternary solid-solution models,involving any other pair among the six Eu end-members listed in Table 5, fail to explain all the data,regardless of the assumed Go

f values.

pH 6

pH 8

# 1

model

cEu

0.001 0.01 0.1 110-7 10-6 10-5 10-4

10-7

10-8

10-9

10-10

10-11

10-12

10-13

10-6

mE

u

# 2

data

pH 13

Eu(OH) 3

Eu (CO ) 2 3 3

Eu(OH)CO 3

Fig. 8:Experimental results (symbols) comparedwith predicted isotherms (lines) based on thesame ternary EuH(CO3)2 – Eu(OH)3 – CaCO3ideal solid solution model. Horizontal linesdefine the solubility limits set by pure Eu-solids under the relevant experimentalconditions.

In conclusion, the GEM thermodynamic calculationswe have performed suggest that heterovalentcarbonate solid solutions are complex, and mayinvolve incorporation of multiple trace-metal species.The identification of appropriate end-members is nottrivial, and requires, firstly, extensive partitioningsolubility data to be obtained, which should then beprocessed using the GEM DualTh technique, and,secondly, be supported by spectroscopic evidence.In this particular case, our ternary solid solutionmodel is corroborated by independent laserfluorescence data [31], indicating the existence of twoindependent CmIII species in calcite, one beingpartially hydrated.

7 ON-GOING AND FUTURE DEVELOPMENTSContinuous development and support of the GEMS-PSI code is aimed at improving quality andperformance, first of all by adding the newfunctionality important for performance assessmentof nuclear waste repositories. At the same time, thisimprovement will be of benefit for a growing scientificcommunity of GEMS users. Notwithstanding theroutine IT-engineering tasks (aimed at optimisingresearchers’ working time), three lines ofdevelopment, considered to be important in the nearfuture, should be mentioned. These are itemised inthe sub-sections below.

7.1 Sorption Continuum and the D u a l T hModule

The GEMS-PSI code offers two alternativeapproaches for the thermodynamic modelling of theuptake by mineral solids of dissolved elements: thesurface complexation models (SCM) [14,32], and thesolid–solution aqueous-solution (SSAS) models (seeexamples above). Taken together or alone, theseapproaches can generate “smart Kd” values, bymodelling the whole ”sorption continuum” (fromadsorption to solid solution) in elementalstoichiometry only, without additional balanceconstraints for surface sites. This strategy can help inthe prediction of stability of clay minerals, and theradionuclide sorption in them, over long time-scales[33]. It also opens a way towards future developmentof a uniform chemical thermodynamic database for(trace) element sorption [32] as a consistentextension of the existing databases for aqueousspecies, gases and minerals.The DualTh techniques (see Sections 3.2 and 6) canhelp considerably in interpreting Kd values from

121

sorption experiments, in terms of standardthermodynamic properties of solid-solution end-members or surface complexes. Given that the uvector is computed by GEMS from each experimentalaqueous composition, the remaining calculations aresimple, and can be performed using, for example, anExcel spreadsheet. However, this is a rather time-consuming process, because of the need to exportGEM results and related thermodynamic data fromthe GEMS database to the spreadsheet, and thenimport the DualTh estimated values back into theGEMS modelling project, and the fact that the entirecycle is usually repeated several times.Many of these tedious operations could be avoided ifDualTh calculations were implemented as a separatemodule in the GEMS package, and as one having adirect access to the necessary parts of the projectdatabase. This would save up to 90% of theresearcher’s time now needed for preparing theDualTh spreadsheet. Thus, implementation of theDualTh module is a top priority task in the furtherdevelopment of the GEMS-PSI code.7.2 Model Sensitivity and the UnSpace ModuleIn geochemical modelling, the input data are usuallytreated in a deterministic way, though, in reality,neither the thermodynamic data ( o

TjG ,,gj ) nor the

variables of state (b , P, T) are known precisely.Consequently, by taking all input data asdeterministic, the modeller risks being led to wrongconclusions. When reasonable uncertainty intervalsare available from analytical error estimates, or froma critical compilation of thermodynamic constants, thesensitivity and robustness of a particular geochemicalmodel can be assessed using Monte Carlosimulations, which generate many (hundreds tothousands) of “sample calculations”, and treat themstatistically.We believe that the main difficulty in these kinds ofstudies lies in the interpretation of sampled outputdata. As shown by Karpov and his co-workers [34],for sufficiently large uncertainties in input data, the apriori correct result may not be the most frequent oneamong the generated sample variants.To solve the problem, a new method has beenproposed that involves: (i) the GEM IPM algorithmused for sampling the multi-dimensional “uncertaintyspace” over a uniform probing grid; and (ii) an advan-ced analysis, combining “dual thermodynamics” withthe “decision-making” criteria projected from gametheory. In a pilot study [35], performed in colla-boration with Prof. Karpov’s group in Russia, thepotential of this method has been explored. We havealso found that independent experimental informationcan be used in combination with decision-makingcriteria to “filter” the “uncertainty space”. This helps indiscarding unrealistic sample variants, which furtherreduces the initially-applied uncertainty intervals. We

believe that, in this way, a much more robustinterpretation of thermodynamic modelling results canbe finally achieved.The “uncertainty space” concept, as outlined above,opens an exciting research perspective. As asupporting tool for future model sensitivity studies, anew UnSpace module will be implemented into aforthcoming version of the GEMS-PSI code.

7.3 Coupling with Fluid-Mass-Transport CodesPrediction of radionuclide migration implies simul-taneous accounting of the diffusive/advective masstransport and the chemical interactions. The lattermay cause dissolution/precipitation of the solidphases, which, in turn, may influence porosity,hydraulic conductivity and mass-transport pathways[36].In principle, the calculation of equilibrium states inGEM proceeds without considering chemicalreactions and depends only on changing the bulkcomposition b, T and P of the chemically reactivesub-system. Hence, it is possible to couple GEM witha fluid-mass-transport (FMT) algorithm in a “cleanlyseparated”, modular-based fashion; i.e. withoutintroducing any chemical mass-action terms into themass-transport equations. In such coupling schemes,the FMT part should take care only of mass/heatconservation across the node borders, while theGEM algorithm would fully account for thelocal/partial equilibration in each nodal volume.In the foreseen implementation, the first step wouldbe to develop the necessary data structures for theinformation exchange between the FMT and GEMparts. The second step would then be to isolate theIPM-2 algorithm into a separate “kernel” module,which could communicate via those data structureswith any FMT code (and with the GEMS-PSI GUIshell); the GEM IPM “kernel” module should also beable to run on parallel computers. Thus, with such astrategy implemented, it would be possible toproduce a new generation of FMT-GEM coupledalgorithms and codes for modelling, for example,near- and far-field repository systems, with the mostextensive account being taken of the chemicalinteractions and induced FMT parameter changes.

ACKNOWLEDGMENTSWe are greatly indebted to other members of GEMS-PSI development team: namely, K.V. Chudnenko,S.V. Dmitrieva, I.K. Karpov, A.V. Rysin, andT.!Thoenen, who helped make the GEMS package areality. The work presented in this article was fundedin part by the Swiss National Cooperative for theDisposal of Radioactive Waste (NAGRA).

122

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