Comprehensive nuclear materials 1 17 computational thermodynamic application to nuclear materials

Comprehensive nuclear materials 1 17 computational thermodynamic application to nuclear materials Comprehensive nuclear materials 1 17 computational thermodynamic application to nuclear materials Comprehensive nuclear materials 1 17 computational thermodynamic application to nuclear materials Comprehensive nuclear materials 1 17 computational thermodynamic application to nuclear materials Comprehensive nuclear materials 1 17 computational thermodynamic application to nuclear materials

1.17 Computational Thermodynamics: Application to Nuclear Materials T M Besmann Oak Ridge National Laboratory, Oak Ridge, TN, USA Published by Elsevier Ltd 1.17.1 1.17.2 1.17.3 1.17.4 1.17.4.1 1.17.4.2 1.17.4.3 1.17.4.4 1.17.4.5 1.17.4.6 1.17.5 1.17.6 1.17.7 References Introduction Thermochemical Principles The CALPHAD Approach and Free Energy Minimization Treatment of Solutions Regular Solution Models Variable Stoichiometry/Associate Species Models Compound Energy Formalism Thermochemical Modeling of Defects Modified Associate Species Model for Liquids Ionic Sublattice/Modified Quasichemical Model for Liquids Thermochemical Data Sources Thermochemical Equilibrium Computational Codes Outlook Abbreviations CALPHAD CEF DTA EMF NASA NIST MOX TRU Calculation of phase diagrams Compound energy formalism Differential thermal analysis Electro-motive force National Aeronautics and Space Administration National Institute of Standards and Technology Mixed oxide fuel Transuranic Symbols Cp E EBW Eij EQM G Gex Gid H Hmix Heat capacity at constant pressure Energy of the system Bragg–Williams model energetic parameter Interaction energy between components i and j Quasichemical model energetic parameter Gibbs free energy Excess free energy Free energy contribution due to ideal entropy of mixing Heat or enthalpy Heat of mixing L n P pO2* R S s Sconfig T m V X y z Z 455 456 457 457 459 460 461 462 463 465 467 468 468 469 Interaction parameter, typically of the form aỵbT Moles of a constituent Pressure A dimensionless quantity defined by the oxygen pressure divided by the standard state pressure Ideal gas law constant Entropy Index for a sublattice Configurational entropy Absolute temperature Chemical potential Volume Mole fraction The site fraction for species j Stoichiometric coefficient Nearest neighbor coordination number 1.17.1 Introduction Nuclear fuels and structural materials are complex systems that have been very difficult to understand and model despite decades of concerted effort Even single actinide oxide or metallic alloy fuel forms have yet to be accurately, fully represented The problem is 455 456 Computational Thermodynamics: Application to Nuclear Materials compounded in fuels with multiple actinides such as the transuranic (TRU) fuels envisioned for consuming long-lived isotopes in thermal or fast reactors Moreover, a fuel that has experienced significant burnup becomes a very complex, multicomponent, multiphase system containing more than 60 elements Thus, in an operating reactor the nuclear fuel is a high-temperature system that is continuously changing as fission products are created and actinides consumed and is also experiencing temperature and composition gradients while simultaneously subjected to a severe radiation field Although structural materials for nuclear reactors are certainly complex systems that benefit from thermochemical insight, the emphasis and examples in this chapter focus on fuel materials for the reasons noted above The higher temperatures of fuels quickly drive them to the thermochemical equilibrium state, at least locally, and their compositional complexity benefits from computational thermochemical analysis Related information on thermodynamic models of alloys can be found in Chapter 2.01, The Actinides Elements: Properties and Characteristics; Chapter 2.07, Zirconium Alloys: Properties and Characteristics; Chapter 2.08, Nickel Alloys: Properties and Characteristics; Chapter 2.09, Properties of Austenitic Steels for Nuclear Reactor Applications; Chapter 1.18, Radiation-Induced Segregation; and Chapter 3.01, Metal Fuel A major issue for nuclear fuels is that the original fuel material, whether the fluorite-structure phase for oxide fuels or the alloy for metallic fuels, has variable initial composition and also dissolves significant bred actinides and fission products Thus, the fuel phase is a complex system even before irradiation and becomes significantly more complex as other elements are generated and dissolve in the crystal structure Compounding the complexity is that, after significant burnup, sufficient concentrations of fission products are formed to produce secondary phases, for example, the five-metal white phase (molybdenum, rhodium, palladium, ruthenium, and technicium) and perovskite phases in oxide fuels as described in detail in Chapter 2.20, Fission Product Chemistry in Oxide Fuels Thus, any chemical thermodynamic representation of the fuel must include models for the nonstoichiometry in the fuel phase, dissolution of other elements, and formation of secondary, equally complex phases Dealing with the daunting problem of modeling nuclear fuels begins with developing a chemical thermodynamic (or thermochemical) understanding of the material system Equilibrium thermodynamic states are inherently time independent, with the equilibrium state being that of the lowest total energy Therefore, issues such as kinetics and mass transport are not directly considered Although the chemical kinetics of interactions are important, they are often less so in the fuel undergoing burnup (fissioning) because of the high temperatures involved and resulting rapid kinetics and can often be neglected on the time scales involved for the fuel in reactor The time dependence of mass transport, however, does influence fuel behavior as evidenced by the significant compositional gradients found in high burnup fuel whether metal or oxide, and most notably by attack of the clad by fission products and oxidation by species released from oxide fuel Although the equilibrium state provides no information on diffusion or vapor phase transport, it does provide source and sink terms for these phenomena Thus, the calculation of local equilibrium within fuel volume elements can in principle provide activity/ vapor pressure values useful in codes for computing mass flux Thermochemically derived properties of fuel phases also provide inherent thermal conductivity, source terms for grain growth, potential corrosion mechanisms, and gas species pressures, all important for fuel processing and in-reactor behavior Thermochemical insights can therefore provide support for modeling species and thermal transport in fuels 1.17.2 Thermochemical Principles Understanding the chemical thermodynamic behavior of reactor materials means describing multicomponent systems with regard to their relative free energies For nuclear fuels that includes both stoichiometric phases as well as solid and liquid solutions containing multiple elements and the vapor species they generate The total free energy determined from the thermochemical descriptions for all the potential phases is computed, and those phases/compositions that result in the lowest free energy state represent the equilibrium system The expression of the free energy is in terms of the Gibbs free energy, G, at constant temperature and pressure, following the familiar relation G ẳ E ỵ PV TS ẵ1 where E is the energy of the system, P is pressure, V is volume, T is absolute temperature, and S is entropy A convenient expression at equilibrium in a constant temperature and pressure system is G ẳ H TS ẵ2 Computational Thermodynamics: Application to Nuclear Materials where H is the heat or enthalpy The temperature dependence of the enthalpy is related to heat capacity, Cp , by ðT H ¼ H298 ỵ Cp dT ẵ3 Cp =T dT ẵ4 298 and to entropy by T S ẳ S298 ỵ 298 The temperature dependence for Cp is expressed as a polynomial from which it is possible to generate what is termed the Gibbs free energy function, which is usually expressed as G ẳ A ẳ BT ỵ CT lnT ỵ DT ỵ ET ỵ F =T ẵ5 The Gibbs free energy function is a very convenient form to work with, particularly for free energy minimization software that computes an equilibrium state That is defined as Gibbs free energy of a system that is at its minimum value, or @G ¼ A very useful value to use when working with complex systems is the chemical potential, m, which is the partial derivative of the Gibbs free energy with respect to the moles or mole fraction of a constituent Thus, at constant temperature and pressure mi ẳ @G=@ni ịT ;P ½6Š where n is the number of moles of the constituent For constant temperature and pressure X mi dni ½7Š @G ¼ i A system’s equilibrium state is therefore computed by minimizing the total free energy expressed as the sum of the various Gibbs free energy functions constrained by the mass balance with a resulting assemblage of phases and their amounts 1.17.3 The CALPHAD Approach and Free Energy Minimization The overall development of a consistent thermochemical representation for the phase equilibria and thermodynamics of a system utilizing all available information has been termed the CALPHAD (computer coupling of phase diagrams and thermochemistry) approach.1 Whether free energy and heat 457 capacity data are provided from first principles calculations or experimentally, for example, from differential scanning calorimetry, solution calorimetry, or thermogravimetric measurements, is irrelevant as long as the information is accurate and applicable The situation is similar for phase equilibria, that is, what phases form under what conditions The developed phase diagrams provide information that can be used to fit prospective thermochemical models This data, together with current computational methods that facilitate development of accurate representations for systems reproducing observed behavior, define the CALPHAD methodology The results ideally are databases for specific components that may also be used in the construction of systems with yet larger numbers of constituents A schematic of the CALPHAD approach can be seen in Figure The CALPHAD approach assumes that the systems being assessed are in equilibrium, that is, the lowest energy state under given conditions of temperature, pressure, and composition The previous section describes the mathematical relationships that govern minimization of the total free energy Traditionally, one determined the minimum free energy state by writing competing reactions related with equilibrium constants, with the phase assemblage from the reaction that yielded the most negative Gibbs free energy state being the most stable.3,4 A more generalized approach was developed in the 1950s by White et al.5 using Lagrangian multipliers Zeleznik and Gordon6 investigated the major approaches to computing equilibrium states, which led to their development of a computer code for computing equilibrium at NASA The techniques were further developed by van Zeggeren and Storey7,8 through the 1960s Ultimately, Eriksson9–11 developed an approach that was generally applicable to a wide variety of systems and included solution phases that could be nonideal This led to the widely used code SOLGASMIX,11 whose equilibrium calculational methodology remains central to many contemporary software packages While SOLGASMIX appears to be the first, other codes for equilibrium calculations such as those noted in Section 1.17.6 had similar developmental histories 1.17.4 Treatment of Solutions Whether it is the nonstoichiometry of fluoritestructure UO2Ỉx or variable composition orthorhombic or tetragonal U–Zr alloy fuel, the accurate thermochemical description of these phases has 458 Computational Thermodynamics: Application to Nuclear Materials Ab-initio calculations Thermodynamic optimization Theory quantum mechanics statistical thermodynamics Estimates Experiments DTA, calorimetry, EMF, vapor pressure metallography, X-ray diffraction, Models with adjustable parameters Adjusting the parameters Thermodynamic functions G, H, S, Cp = ¦ (T, P, X, ) Storage databases, publications Equilibrium calculations Equilibria Graphical representation Phase diagrams Applications Figure Diagram illustrating the computer coupling of phase diagrams and thermochemistry approach after Zinkevich.2 been through the use of solution models Solid and liquid solution modeling from simple highly dilute systems to more complex interstitial and substitutional solutions with multiple lattices has been a rich field for some time, yet it is far from fully developed To accurately describe the energetics of solutions will eventually require bridging atomistic models with the mesoscale treatments currently being used Recent approaches have begun to deal with defect structures in phases, although only in a very constrained manner which limits clustering and other phenomena Yet, when they are coupled with accurate data that allow fitting of the model parameters, the resulting representations have been highly predictive of phase relations and chemistry A number of texts provide useful descriptions of solution models from the simple to the complex.12–15 While there are several relatively accurate but rather intricate approaches, such as the cluster variation method, the discussion in this chapter is confined to simpler models that are easily used in thermochemical equilibrium computational software and thus applicable to large, multicomponent systems of interest in nuclear fuels The simplest model is the ideal solution where the constituents are assumed to mix randomly with no structural constraints and no interactions (bonding or short- or long-range order) The standard Gibbs free energy and ideal mixing entropy contributions are X   j ni Gi G ¼ X G id ẳ RT ni ln ni ẵ8  where G is the weighted sum of the standard Gibbs free energy for the constituents in the j phase solution, Gid is the contribution from the entropy increase due to randomly mixing the constituents, which is the configurational entropy, and R is the ideal gas law constant For an ideal solution, the sum of the two represents the Gibbs free energy of the system Computational Thermodynamics: Application to Nuclear Materials where L is the coefficient of the expansion in k, which can also have a temperature dependence typically of the form a ỵ bT Thus, a regular solution is defined as k equals zero leaving a single energetic term This approach is related to the Bragg–Williams description, with random mixing of constituents yet with enthalpic energetic terms such that In cases where there are significant interactions (bonding or repulsive interaction energies) among mixing constituents, an energetic term or terms need to be added to the solution free energy The inclusion of a simple compositionally weighted excess energy term, G ex, accounts for the additional solution energy for what is historically termed a regular solution XX Xi Xj Eij ½9Š G ex ¼ G ex ¼ XA XB EBW i¼1 j >1  1.17.4.1 ½10Š Regular Solution Models A common formalism for excess energy expressions is the Redlich–Kister–Muggianu relation, which for a binary system can be written as X Lk;ij ðXi À Xj ịk ẵ11 G ex ẳ Xi Xj k 2500 2500 Model of Gürler et al Model of Jacob et al 2236 Liquid 2000 2000 Temperature (K) 1827 1669 XRh = 0.33 1500 1500 Solid (fcc) 1210 XRh = 0.55 1183 Pd (fcc) + Rh (fcc) 1000 1000 0.0 Pd 0.2 ½12Š Here, XAXB represents a random mixture of A and B components and is thus the probability that A–B is a nearest-neighbor pair, and EBW is the Bragg– Williams model energetic parameter In a relevant example, Kaye et al.16 have generated a solution model for the five-metal white phase noted above and more extensively discussed in Chapter 2.20, Fission Product Chemistry in Oxide Fuels A binary constituent of the model is the fcc-structure Pd–Rh system, which at elevated temperatures forms a single solid solution across the entire compositional range The phase diagram of Figure also shows a lowtemperature miscibility gap, that is, two coexisting identically structured phases rich in either end member The excess Gibbs free energy expression for where X is the mole fraction and Eij the interaction energy between the components i and j The system free energy is thus G ẳ G ỵ G id þ G ex 459 0.4 0.6 0.8 XRh 1.0 Rh 16 Figure Computed Pd–Rh phase diagram with indicated data of Kaye et al illustrating complete fcc solid-solution range Reproduced from Kaye, M H.; Lewis, B J.; Thompson, W T J Nucl Mater 2007, 366, 8–27 from High Temperature Materials Laboratory 460 Computational Thermodynamics: Application to Nuclear Materials -4 the fcc phase was determined from an optimization using tabulated thermochemical information together with the phase equilibria and yielded -8 ½13Š 1.17.4.2 Variable Stoichiometry/Associate Species Models As noted in Chapter 2.01, The Actinides Elements: Properties and Characteristics; Chapter 2.02, Thermodynamic and Thermophysical Properties of the Actinide Oxides; and Chapter 2.20, Fission Product Chemistry in Oxide Fuels, modeling of complex systems such as U–Pu–Zr and (U,Pu)O2Ỉx has been exceptionally difficult For example, actinide oxide fuel is understood to be nonstoichiometric almost exclusively due to oxygen site vacancies and interstitials As a result, the fluoritestructure phase has been treated as being composed of various metal-oxygen species with no vacancies on the metal lattice An early and successful modeling approach has employed a largely empirical use of variable stoichiometry species that are mixed as subregular solutions to fit experimental information.17–20 This technique can be viewed as a variant of the associate species method.21 In the approach, thermochemical values were determined from the phase equilibria, that is, the phase boundaries, and data for the temperature– composition–oxygen potential [mO2 ¼ RT ln(pO2*)] where pO*2 is a dimensionless quantity defined by the oxygen pressure divided by the standard state pressure of bar The UO2Ỉx phase, for example, was treated as a solid solution of UO2 and UaOb where the values of a and b were determined by a fit to experimental data Figure illustrates the trial and error process using a limited data set to obtain the species stoichiometry which results in the best fit to the data As can be seen, a variety of stoichiometries for the constituent species yield differing curves of ln(pO2*) versus f(x), with the most appropriate matching the slope of Thus, for this example U10/3O23/3 provides for an optimum fit between U3O7 and U4O9, and its solution with UO2 best reproduces the observed oxygen potential behavior Utilizing a much more extensive data set from a variety of sources resulted in a set of best fits to the data, yet they required three solid solutions to adequately represent the entire compositional range for UO2Ỉx These are -12 In (pO2* ) G ex ẳ XPd XRh ẵ21247 ỵ 2199XRh 2:74 À 0:56XRh ÞT Š -16 Raw data Hagemark and Broli, 1673 K Roberts and Walter, 1695 K [UO2] + [UO3] [UO2] + [U2O5] [UO2] + [U3O7] -20 [UO2] + [U10/3 O23/3] [UO2] + [U4O9] Slope -2 -24 -8 -4 f (X) Figure The ln(pO2* ) dependence as a function of x for UO2ỵx and of f(x) for several solid-solution species’ stoichiometries for an illustrative oxygen pressure– temperature–composition data set Coincidence with the theoretical slope of indicates the proper solution model Reproduced from Lindemer, T B.; Besmann, T M J Nucl Mater 1985, 130, 473488 UO2ỵx (high hyperstoichiometry, i.e., large values of x): UO2 ỵ U3O7, UO2ỵx (low hyperstoichiometry, i.e., smaller values of x): UO2 ỵ U2O4.5, and UO2x (hypostoichiometric): UO2 ỵ U1/3 The results of the models for UO2Ỉx are plotted in Figure together with the entire data set used for optimizing the system The above models for UO2Ỉx have been widely adopted, as has been a similar model of PuO2Àx 16 These have also been combined to construct a successful model for (U,Pu)O2Ỉx 16 Lewis et al.22 used an analogous technique for UO2Ỉx Lindemer23 and Runevall et al.24 have generated successful models of CeO2Àx Runevall et al.24 also used the method for NpO2Àx , AmO2Àx (with the work of Thiriet and Konings25), (U,Am)O2Ỉx, (Th,U)O2Ỉx, (U,Ce)O2Àx, (Pu,Am)O2Ỉx, and (U,Pu,Am)O2Ỉx They noted that results for the (Th,U)O2ỵx were less successful perhaps because of the difficulty in the measurements Computational Thermodynamics: Application to Nuclear Materials ) UO ( n xi x 2+ 0.2 25 0.2 461 U–O liquid region 0.15 0.1 0.03 0.01 0.006 -200 0.0 0.0 06 03 0.0 10-3 -400 0.0 10 -4 10 -3 Oxygen potential (kJ mol–1) 0.003 10 - 10 -5 10 -6 U–O liquid region (UO 2-x ) X = 0.3 x in (U O 2) ex ac t 0.2 10 - -600 10 -4 0.1 -800 500 1000 1500 2000 2500 3000 Temperature (K) Figure Oxygen potential plotted versus x for the models of UO2Ỉx of Lindemer and Besmann17 overlaid with the entire data set used for the optimization made near stoichiometry Osaka et al.26–28 used the approach to successfully represent the (U,Am)O2Àx, (U,Pu,Am)O2Àx, and (Am,Th)O2Àx phases 1.17.4.3 Compound Energy Formalism Regular or subregular solution and variable stoichiometry representations, while relatively successful, lack a sense of reproducing physical processes Specifically, they are constrained with regard to accurately dealing with entropy contributions because of the defect structures in nonstoichiometric phases and substitutional solutions A practical advance has been the sublattice approach, which has been further refined for crystalline systems in the compound energy formalism (CEF).29 As typical for cation– anion systems, the structure of a phase can be represented by a formula, for example, (A,B)k(D,E)l where A and B mix on one sublattice and D and E mix on a second sublattice The constitution of the phase is made up of occupied site fractions, and allowing one of the constituents to be a vacancy permits treatment of nonstoichiometric systems Even with a sublattice approach such as CEF, the relationship of eqn [10] is still applicable, but with an interpretation related to a sublattice model The sum 462 Computational Thermodynamics: Application to Nuclear Materials of the standard Gibbs free energies in this case is the sum of the values for the paired sublattice constituents, which for the example above might be AkDl Each is a unique set with the Gibbs free energies for the constituents derived from the end-member standard Gibbs free energies, typically through simple geometric additions with any necessary additional configurational entropy contributions The entropy contribution from mixing on the sublattice sites is defined as X zs y s lnyjs ị ẵ14 G id ẳ RT where z is the stoichiometric coefficient, s defines the lattice, and y is the site fraction for species j Excess terms represent the interaction energetics between each set of sublattice constituents, for example, AkDl : BkDl Again, a Redlich–Kister–Muggianu formulation that includes expansion terms for interactions between the constituents can be used: XXX yi1 yj2 yk1 Li; j :k G xs ¼ i þ j k XXX i j k yk2 yi1 yj1 Lk:i; j ½15Š where the sums are associated with components on each sublattice and and the L values are terms for the interaction energies between cations i and j on one sublattice when the other sublattice is occupied only by cation k, and vice versa for the second term The PuO2Àx phase has been successfully represented by a CEF approach by Gueneau et al.30 The phase can be described by two sublattices with vacancies only on the anion sites (Pu4ỵ,Pu3ỵ)1(O2,Va)2 Including the end members, the constituent species are then (Pu4ỵ)1(O2)2, (Pu4ỵ)1(Va)2, (Pu3ỵ)1(O2)2, and (Pu3ỵ)1(Va)2 A schematic of the relationship between the constituents is seen in Figure where the corners represent each of the constituents listed above The charged constituents must sum to neutrality, and the line designating neutrality is seen in Figure Gibbs free energy expressions for each of the units can be determined from standard state values Optimizations using all available thermochemical information, for example, oxygen potentials and phase equilibria, can thus yield the necessary corrections to the Gibbs free energies for the nonstandard constituents together with obtained interaction parameters (L values) The results are shown in Figure where oxygen potential isotherms overlay the phase diagram and which shows mO2 results of models for other phases in the system The CEF approach has recently begun to be more widely applied to nuclear fuels Besides the PuO2Àx system noted above, Gueneau et al.31 also applied the model to accurately describe solid solution phases in the U–O system, as has Chevalier et al.32 who also addressed the U–O–Zr ternary system.33 Kinoshita et al.34 used a sublattice approach to model fluoritestructure oxides including ThO2Àx and NpO2Àx, although they did not include charged ionic cations and anions on the sublattices Zinkevich et al.35 successfully modeled the CeO2Àx phase using the CEF approach in their comprehensive assessment of the Ce–O phase diagram 1.17.4.4 Thermochemical Modeling of Defects Another way to view solid solutions and nonstoichiometry is as a function of defects in the ideal lattice (Pu3+)1(Va)2 (+3) PuO1.5 = (Pu3+)1(Va1/4, O2-3/4)2 (0) (Pu3+)1(O2-)2 (-1) (Pu4+)1(Va)2 (+4) Neutral line (Pu4+)1(O2-)2 = PuO2 (0) Figure Compound energy formalism sublattice model illustration of the components and their charge in a two-dimensional representation after Gueneau et al.31 Computational Thermodynamics: Application to Nuclear Materials 463 -4 -8 log10 (pO2) in bar -12 -16 -20 -24 -28 -32 -36 1.5 1.6 1.7 1.8 O/Pu ratio 1.9 2.0 Figure Oxygen potentials overlaying the phase equilibria for the Pu–O system as computed by Gueneau et al.31 showing the results of the fit to the compound energy formalism model and representative data for the PuO2Àx phase Reproduced from Gueneau, C.; Chatillon, C.; Sundman, B J Nucl Mater 2008, 378, 257–272 This has been of particular interest for oxide fuels as they are seen to govern dissolution of cations and nonstoichiometry in oxygen behavior and as a result, transport properties Defect concentrations are inherent in the CEF, as vacancies and interstitials on the oxygen lattice for fluorite-structure actinide systems are treated as constituents linked to cations (see Section 1.17.4.3) A more explicit treatment of oxide systems with point defects has been applied to a wide range of materials such as high-temperature oxide superconductors, TiO2, and ionic conducting membranes, among others For oxide fuels, point defects have been described thermochemically by a number of investigators starting as early as 1965 with more recent treatments in fuels by Nakamura and Fujino,36 Stan et al.,37 and Nerikar et al.38 Oxygen site defects, which dominate in the fluorite-structure fuels, are of course driven by the multiple possible valence states of the actinides, most notably uranium, which can exhibit Uỵ3, Uỵ4, Uỵ5, and Uỵ6 A simple example of the point defect treatment can be seen in Stan et al.37 They optimized defect concentrations from the defect reactions described in the Kroger–Vinck notation 00 OO ẳ Oi ỵ VO 00 O2 ỵ 2U U ẳ Oi ỵ 2UU A dilute defect concentration was assumed such that there were no interactions between defects and thus no excess energy terms The results of the fit to literature data are seen in Figure 7(a), where the stoichiometry of the fluorite-structure hyperstoichiometric urania is plotted as a function of defect concentration xa The relationships were also used to compute oxygen potentials as a function of stoichiometry and are plotted in Figure 7(b) illustrating relatively good agreement with values computed by Nakamura and Fujino.36 1.17.4.5 Modified Associate Species Model for Liquids The liquid phases in nuclear fuels are important to model so that the phase equilibria can be completely assessed through comparison of experimental and computed phase diagrams The availability of solidus and liquidus information also provides necessary boundaries for modeling the solid-state behavior Finally, safety analysis requirements with regard to the potential onset of melting will benefit from accurate representations of the complex liquids 464 Computational Thermodynamics: Application to Nuclear Materials 0.14 0.25 0.12 Oi 1373 K •• VO 0.15 U• x in UO2+x Xa 0.2 Oi U 0.1 0.10 0.05 V• • O 0.15 0.1 x in UO2+x 0.05 (a) 1273 K 1373 K 0.06 Model Model 0.02 0.00 -12 Nakamura and Fujino 0.08 0.04 1273 K UO2+x (b) -10 -8 log10 pO2 (atm) -6 Figure (a) Concentrations of defect species in UO2ỵx relative to the concentration of oxygen sites in the perfect lattice, as a function of nonstoichiometry, calculated with a defect model (b) UO2ỵx nonstoichiometry as a function of partial pressure of oxygen (Dashed line is model-derived and solid line are results of Nakamura and Fujino36 and Stan et al.37) Ideal, regular/subregular, or Bragg–Williams formulations are not very successful in representing metal and especially oxide liquids where there are strong interactions between constituents The CEF model is designed for fixed lattice sites, and thus it too will not handle liquids The issues for these complex liquids involve the short-range ordering that generally occurs and its effect on the form of the Gibbs free energy expressions One approach to dealing with the issue of these strong interactions is the modified associate species method The modified associate species technique for crystalline materials was discussed to an extent in Section 1.17.4.2 Its application to, for example, oxide melts has been more broadly covered recently by Besmann and Spear39 with much of the original development by Hastie and coworkers.40–43 The approach assumes that the liquid can be modeled by an ideal solution of end-member species together with intermediate species The modified term refers to the fact that an ideal solution cannot represent a miscibility gap in the liquid as that requires repulsive (positive) interaction energy terms Thus, when a miscibility gap needs to be included, interaction energies between appropriate associate species are added to the formulation In the associate species approach, the system standard Gibbs free energy is simply the sum of the constituent end-member and associate free energies, for example, A, B, and A2B, where inclusion of the A2B associate is found to reproduce the behavior well,     G ¼ XA GA ỵ XB GB ỵ XA2 B GA2 B ẵ16 Consequently, ideal mixing among end members and associates generates the entropy contribution G id ẳ RT XA ln XA ỵ XB ln XB ỵ XA2 B ln XA2 B ị ½17Š Should a nonideal term providing positive interaction energies be needed to properly address a miscibility gap, it would be added into the total Gibbs free energy as in eqn [10] For example, for an interaction between A and A2B in the Redlich–Kister–Muggianu formulation the excess term is expressed as X Lk;A:A2 B XA XA2 B ịk ẵ18 G ex ¼ XA XA2 B k The associate species are typically selected from the stoichiometry of intermediate crystalline phases, but others as needed can be added to accurately reflect the phase equilibria even when no stable crystalline phases of that stoichiometry exist Gibbs free energies for these species can be derived from fits to the phase equilibria and other data following the CALPHAD method with first estimates generated from crystalline phases of the same stoichiometry or weighted sums of existing phases when no stoichiometric phase exists The application of the method for the liquid phase in the Na2O–Al2O3 is seen in the computed phase diagram in Figure For this system, the associate species required to represent the liquid were only Na2O, NaAlO2, (1/3) Na2Al4O7, and Al2O3 In nuclear fuel systems, Chevalier et al.44 applied an associate species approach using the components O, U, and O2U, although it deviated from the associate species approach in using binary interaction parameters in a Redlich–Kister–Muggianu form The computed Computational Thermodynamics: Application to Nuclear Materials 465 2200 0.87 (0.89) Liquid 2000 1869 (1867) 2054 (2054) 1885 (1976) Liquid + NaAIO2 Liquid + b-alumina 1600 1443 (1443) 1200 NaAlO2 1400 0.01 1126 1000 0.0 Na2O 0.2 0.4 0.6 Mole fraction Al2O3 0.8 NaAl9O14 (b-alumina) 1584 0.68 (1585) (0.63) Na2Al12O18 (bЈЈ-alumina) Temperature (º C) 1800 1.0 Al2O3 Figure Calculated phase diagram for the Na2O–Al2O3 system using the modified associate species approach for the liquid Values in parentheses are the accepted phase equilibria temperatures or compositions shown for comparison with the results of the modeling Reproduced from Chevalier, P Y.; Fischer, E.; Cheynet, B J Nucl Mater 2002, 303, 1–28 phase diagram showing agreement with the liquidus/ solidus data is seen in Figure The use of the modified associate species model with ternary and higher order systems can require the use of ternary or possibly quaternary associates Another issue with the modified associate species approach is that in the case of a highly ordered solution which requires an overwhelming content of an associate compared to an end-member, the relations not follow what should be Raoult’s law for dilute solutions At the other extreme, it is also apparent that in the case of essentially zero concentration of associates, the relationships not default to an ideal solution as one would expect 1.17.4.6 Ionic Sublattice/Modified Quasichemical Model for Liquids In contrast to using associates for liquid solutions is a sublattice approach in which cations and anions are mixed on respective lattice sites With anions and cations assigned to specific sublattices, it is possible to capture interactions and short-range order with species occupying the sites and additional energetic terms The components can essentially be allowed to independently mix on each sublattice within the energetic constraints and the system free energy minimized.46 The approach has been successfully used by Gueneau et al.47 to model the liquid in the U–O and Pu–O systems where ionic metal species reside on one lattice and oxygen anions, neutral UO2 or PuO2, charged vacancies, and O species on the other An improvement to the simple sublattice approach is the quasichemical method introduced by Fowler and Guggenheim48 and later further developed by Pelton and coworkers.49–52 It approaches short-range order in liquids through the formation of nearest-neighbor pairs on a quasilattice It thus differs significantly from the modified associate species approach such that in the quasichemical method short-range order is accommodated by components pairing and the energetically described extent of like and unlike components pairing The technique thus avoids the paradox where a high 466 Computational Thermodynamics: Application to Nuclear Materials T (K) 3600 L1 3200 L2 a a 2800 fcc C 2400 UO2+5x 2000 O8U3 (s) 1600 1200 U1 (bcc A2) U1 (TET) 800 O9U4 (S) O3U1 (s) 400 0.0 U1 (ORT A20) & 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 x (mol) Figure The computed U–O phase diagram of Chevalier et al.44 for which the liquid was modeled using the associate species approach with selected experimental points indicated Reproduced from Chevalier, P Y.; Fischer, E.; Cheynet, B J Nucl Mater 2002, 303, 1–28 degree of short-range order in the associate species approach causes minimal end-member species content and therefore fails in the limit to be a Raoult’s law solution.52 In the modified quasichemical approach for a simple binary A–B system, the components are treated as distributed on a quasilattice and that an energetic term governs exchange among the pairs A Aị ỵ B Bị ẳ 2A Bị ẵ19 A parameter, Z, represents the nearest-neighbor coordination number such that each component forms Z pairs For one mole of solution, ZXA ẳ 2nAA ỵ 2nAB ẵ20 ZXB ẳ 2nBB ỵ 2nAB ẵ21 where the moles of each pair are nAA, nBB, and nAB The relative proportions of each pair is Xij, where Xij ẳ nij =nAA ỵ nBB ỵ nAB ị ẵ22 The configurational entropy contribution is captured from the random distribution of the pairs over the quasilattice positions The result is a heat of mixing of Hmix ẳ XAB =2ịEQM ẵ23 where EQM is the quasichemical model energetic parameter, and a configurational entropy Sconfig ¼ À RðXA ln XA ỵ XB ln XB ị RZ=2ịXAA lnXAA =XA2 ị ỵ XBB lnXBB =XB2 ị ỵ XAB lnXAB =2XA XB ịị ẵ24 Utilizing Gibbs free energy functions for the components and expanding EQM as a polynomial in a scheme for minimizing the system free energy provide a system for optimization of the liquid using known thermochemical values and phase equilibria Issues such as the displacement of the composition of maximum short-range order from 50% composition are dealt with by assuming different coordination numbers for each component Greater accuracy is obtained by inclusion of the Bragg–Williams model, thus incorporating lattice interactions beyond nearest neighbors The modification to the quasichemical model yields Hmix ẳ EQM =2ịXAB ỵ EBW XA XB ½25Š The extension of the modified quasichemical model to ternary systems is directly possible using only binary model parameters An issue for the modified quasichemical model is that it fails at high deviations from random ordering, although that is generally not a problem because immiscibility will occur before the deviations grow too large The model can also predict a large amount Computational Thermodynamics: Application to Nuclear Materials of ordering that can result in a negative configurational entropy, a physical impossibility.53 1.17.5 Thermochemical Data Sources Tabulated thermochemical data have been available from a number of sources for several decades For general substances, the most familiar have been the NIST-JANAF Thermochemical Tables54 and Thermochemical Data of Pure Substances.55 The data are generally given as 298.15 K values, and columns of values such as Gibbs free energy, heat, entropy, and heat capacity are listed incrementally with temperature The NIST-JANAF Thermochemical Tables are also available online through the National Institute for Standards and Technology (NIST) One of the key issues in using thermochemical data is the consistency of the standard states The current commonplace usage is that the standard state is defined as 298.15 K and bar (100 kPa) pressure Small, but potentially important, errors can arise if data with different standard states are combined, for example, values at standard state pressure of atm and of bar are used together Much of the thermochemical data compilations are currently available as computer databases In addition to the NIST-JANAF Thermochemical Tables54 is that of the Scientific Group Thermodata Europe (SGTE),56 which is well-established and has an ongoing program to assess data and add new species and phases The same is true for the databases provided by THERMODATA57 in Grenoble, France, which has compound and solution values Another source is MALT,46 supplied by Kagaku Gijutsu-Sha in Japan, which is more limited than SGTE,56 focusing on data that directly support industry issues There have also been databases developed specifically for nuclear applications including THERMODATA,57 which has databases for both ex-vessel applications, NUCLEA, and for mixed oxide fuel (MOX) Kurata58 has developed a limited thermochemical database focused on metallic fuels A database dedicated to zirconium alloys of interest for nuclear applications called ZIRCOBASE59,60 is available with fully developed representations of a number of zirconiumcontaining binary systems and some ternaries The binaries and ternaries can be combined in generating higher order systems often with reasonably good accuracy An SGTE56 nuclear materials database is also available containing most of the gaseous species and simple compounds of interest An advanced nuclear 467 fuel-specific database initiated by the Commissariat a` l’Energie Atomique, FUELBASE,31 and which is expected to be moved under the auspices of the Nuclear Energy Agency with the Organization for Economic Cooperation and Development, is described in more detail in Chapter 2.02, Thermodynamic and Thermophysical Properties of the Actinide Oxides Information on the most common compounds and, in recent years, solution phases for many important systems has become available in the literature and is included in databases such as those noted above However, much important data and models are not available for nuclear systems, which have not received the same attention as, for example, commercial steels With advances in first principles modeling, some stoichiometric compounds for which there is limited or no experimental information can have values computationally determined This is more likely for gaseous species than for condensed phases because of the greater ease in modeling the vapor Another approach to filling in needed data is to use simple estimation techniques The heat capacity of a complex oxide can be fairly accurately represented by the linear summation of the values of the constituent oxides A linear relationship with atomic number is often seen in the enthalpy of formation of analogous compounds These and other methods are discussed extensively in Kubaschewski et al.14 Equilibrium computational software packages typically will automatically acquire the needed data from accompanying selected databases The published and commercial databases are generally assessed, meaning that they are compatible with broadly accepted values for the systems and when used with other standard values in the database thus yield correct thermochemical and phase relations However, caution is needed when using those data with additional values obtained from other sources such as published experimental or computed values so that fundamental relationships such as phase equilibria are preserved Another very significant issue is the completeness of the information A simple example is UO2 where calculations can be performed using database values for the phase, whereas in reality the phase varies in stoichiometry as UO2Ỉx and without including a representation for the nonstoichiometry any conclusions will be in doubt Given the great complexity of the fuel and fission product phases described in Chapter 2.01, The Actinides Elements: Properties and Characteristics; Chapter 2.02, Thermodynamic and Thermophysical Properties of the Actinide Oxides; and Chapter 2.20, Fission 468 Computational Thermodynamics: Application to Nuclear Materials Product Chemistry in Oxide Fuels, it is apparent that a thermochemical model of fuel undergoing burnup is far from complete The metallic fuel composition U–Pu–Zr is reasonably well represented,61 largely from the constituent binaries, yet the fuel after significant burnup will also contain bred actinides and fission products Similarly, the oxide fluorite fuel phase with uranium and plutonium has perhaps been completely represented (see Chapter 2.02, Thermodynamic and Thermophysical Properties of the Actinide Oxides), but it too has yet to be modeled containing other TRU elements and fission products High burnup fuels will also generate other phases, as noted in Chapter 2.20, Fission Product Chemistry in Oxide Fuels, and these too are often complex solid solutions with numerous components Thus, the critical question in thermochemical modeling is, does the database contain values and representations for all the species and phases of interest? Without inclusion of all important phases, the accuracy of any conclusions from calculations will be in question As noted above, most databases are assessed, which implies that the included data have been evaluated with regard to the sources and methodologies used to obtain the data It also implies that the data are consistent with information for other phases and species containing one or more of the same components/elements Calculations of properties must return the appropriate relationships between phases and species (e.g., activities and phase equilibria) Thus, the use of data from multiple sources raises the specter of inconsistent values being used, leading to inaccurate representations Assuring that the data are consistent between sources through checks of relationships such as known phase equilibria is important to maintaining confidence in the information providing accurate results 1.17.6 Thermochemical Equilibrium Computational Codes There are a variety of software packages that will perform chemical equilibrium calculations for complex systems such as nuclear fuels These have become quite versatile, able to compute the thermal difference in specific reactions as well as determining global equilibria at uniform temperature or in an adiabatic system They also provide output through internal postprocessors or by exporting to text or spreadsheet applications There are also a variety of output forms including activities/partial pressures, compositions within solution phases, and amounts, which can include plotting of phase and predominance diagrams The commercial products include FactSage62 and ThermoCalc63 which also contain optimization modules that allow use of activity and phase equilibria to obtain thermochemical values and fit to models for solutions Other products include Thermosuite,64 MTDATA,65 PANDAT,66 HSC,67 and MALT.57 1.17.7 Outlook Computational thermodynamics as applied to nuclear materials has already substantially contributed to the development of nuclear materials ranging from oxide and metal fuel processing to assessing clad alloy behavior Yet, in both development of data and models for complex fuel and fuel-fission product systems and in the application of equilibrium calculations to reactor modeling and simulation, there is much to accomplish Databases containing accurate representations of both metallic and oxide fuels with minor actinides are lacking, and even less is known about more advanced fuel concepts such as carbide and nitride fuels Representations for multielement fission products dissolved in fuel phases or as secondary phases generated after considerable burnup are also unavailable, although some simple binary and ternary systems have been determined These are critically needed as they will help govern activities in metal and oxide fuels, influencing thermal conductivity and providing source terms for transport of important species such as those containing iodine The other broad area that needs significant attention is the development of algorithms for computing chemical equilibria Although there are robust and accurate codes for computing equilibria within the software packages discussed in Section 1.17.6, these suffer from relatively slow execution That is not a problem for the codes noted above where only a few calculations are required at any time However, incorporation of equilibrium state calculations in broad fuel modeling and simulation codes with millions of nodes to determine the spatial distribution of phases, solution compositions (e.g., local O/M in oxide fuel), and local activities poses a different problem Current algorithms are far too slow for such use, and therefore, new techniques need to be developed to accomplish these calculations within the larger modeling and simulation codes.68 Computational Thermodynamics: Application to Nuclear Materials Acknowledgments The author wishes to thank Steve Zinkle, Stewart Voit, and Roger Stoller for their valuable 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Zhang, F.; Daniel, S J Miner Met Mater Soc 2003, 55(12), Wu, L.; Themelis, N J CIM Bull 1988, 81(914), 97 Piro, M H.; Welland, M J.; Lewis, B J.; Thompson, W T.; Olander, D R Development of a self-standing numerical tool to compute chemical equilibria in nuclear materials In Proceedings of Top Fuel, Paris, France 2009  ... after Gueneau et al. 31 Computational Thermodynamics: Application to Nuclear Materials 463 -4 -8 log10 (pO2) in bar -12 -16 -20 -24 -28 -32 -36 1. 5 1. 6 1. 7 1. 8 O/Pu ratio 1. 9 2.0 Figure Oxygen... Weinheim, 19 89 470 Computational Thermodynamics: Application to Nuclear Materials 56 Dinsdale, A T CALPHAD 19 91, 15 (4), 317 –425 57 Yokokawa, H.; Yamauchi, S.; Matsumoto, R CALPHAD 2002, 26(2), 15 5 16 2... computed Computational Thermodynamics: Application to Nuclear Materials 465 2200 0.87 (0.89) Liquid 2000 18 69 (18 67) 2054 (2054) 18 85 (19 76) Liquid + NaAIO2 Liquid + b-alumina 16 00 14 43 (14 43) 12 00