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

Nuclear Materials

T M Besmann

Oak Ridge National Laboratory, Oak Ridge, TN, USA

Published by Elsevier Ltd.

1.17.3 The CALPHAD Approach and Free Energy Minimization 457

1.17.4.2 Variable Stoichiometry/Associate Species Models 460

1.17.4.6 Ionic Sublattice/Modified Quasichemical Model for Liquids 465

Abbreviations

CALPHAD Calculation of phase diagrams

CEF Compound energy formalism

DTA Differential thermal analysis

EMF Electro-motive force

NASA National Aeronautics and Space

Administration

NIST National Institute of Standards and

Technology

MOX Mixed oxide fuel

TRU Transuranic

Symbols

C p Heat capacity at constant pressure

E Energy of the system

E BW Bragg–Williams model energetic

parameter

E ij Interaction energy between components

i and j

E QM Quasichemical model energetic parameter

G Gibbs free energy

G ex Excess free energy

G id Free energy contribution due to ideal

entropy of mixing

H Heat or enthalpy

H mix Heat of mixing

L Interaction parameter, typically of the form

a þbT

n Moles of a constituent

pO 2 * A dimensionless quantity defined by the oxygen pressure divided by the standard state pressure

R Ideal gas law constant

s Index for a sublattice

S config Configurational entropy

T Absolute temperature

m Chemical potential

X Mole fraction

y The site fraction for species j

z Stoichiometric coefficient

Z 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

compounded in fuels with multiple actinides such as

the transuranic (TRU) fuels envisioned for consuming

long-lived isotopes in thermal or fast reactors

More-over, a fuel that has experienced significant burnup

becomes a very complex, multicomponent,

multi-phase 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

composi-tion 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

equi-librium state, at least locally, and their compositional

complexity benefits from computational

thermo-chemical analysis Related information on

thermody-namic models of alloys can be found inChapter2.01,

The Actinides Elements: Properties and

Character-istics;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; andChapter3.01, 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

Com-pounding 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 inChapter

2.20, Fission Product Chemistry in Oxide Fuels

Thus, any chemical thermodynamic representation of

the fuel must include models for the

nonstoichiome-try 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 (fis-sioning) 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 infor-mation 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 conductiv-ity, source terms for grain growth, potential corrosion mechanisms, and gas species pressures, all important for fuel processing and in-reactor behavior Thermo-chemical 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 ¼ E þ PV  TS ½1 whereE 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 ¼ H  TS ½2

where H is the heat or enthalpy The temperature

dependence of the enthalpy is related to heat capacity,

Cp, by

H ¼ H298þ

ð

T

298

Cp dT ½3

and to entropy by

S ¼ S298þ

ð

T

298

Cp=T dT ½4

The temperature dependence forCpis 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 ¼ A ¼ BT þ CTlnT þ DT2þ ET3þ F=T ½5

The Gibbs free energy function is a very convenient

form to work with, particularly for free energy

mini-mization 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 0 Avery useful value to

use when working with complex systems is the

chemi-cal 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 ¼ @G=@nð iÞT;P ½6

wheren is the number of moles of the constituent

For constant temperature and pressure

@G ¼X

i

mi dni ½7

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

Free Energy Minimization

The overall development of a consistent

thermo-chemical representation for the phase equilibria and

thermodynamics of a system utilizing all available

information has been termed the CALPHAD

(com-puter coupling of phase diagrams and

thermochem-istry) approach.1 Whether free energy and heat

capacity data are provided from first principles calculations or experimentally, for example, from differential scanning calorimetry, solution calorime-try, 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 devel-oped 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 representa-tions for systems reproducing observed behavior, define the CALPHAD methodology The results ide-ally 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 inFigure 1 The CALPHAD approach assumes that the sys-tems being assessed are in equilibrium, that is, the lowest energy state under given conditions of temper-ature, 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.3,4A more generalized approach was developed in the 1950s by White et al.5

using Lagrangian multipliers Zeleznik and Gordon6 inves-tigated the major approaches to computing equilibrium states, which led to their development of a computer code for computing equilibrium at NASA The tech-niques were further developed by van Zeggeren and Storey7,8through 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,11whose 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 inSection 1.17.6had similar develop-mental histories

1.17.4 Treatment of Solutions

Whether it is the nonstoichiometry of fluorite-structure UO2x or variable composition ortho-rhombic or tetragonal U–Zr alloy fuel, the accurate thermochemical description of these phases has

been through the use of solution models Solid and

liquid solution modeling from simple highly dilute

systems to more complex interstitial and

substitu-tional 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

para-meters, 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 thermochemi-cal equilibrium computational software and thus applicable to large, multicomponent systems of inter-est 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

G¼XniGij

Gid¼ RTXnilnni ½8 whereG

is the weighted sum of the standard Gibbs free energy for the constituents in thej 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

Models with adjustable parameters

Experiments DTA, calorimetry, EMF, vapor pressure metallography, X-ray diffraction,

Theory quantum mechanics statistical thermodynamics

Thermodynamic optimization

Equilibrium calculations

Adjusting the parameters

Equilibria

Phase diagrams representationGraphical

Applications

Thermodynamic functions

G, H, S, C p=¦ (T, P, X, )

Storage databases, publications

Estimates

Figure 1 Diagram illustrating the computer coupling of phase diagrams and thermochemistry approach after Zinkevich 2

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,Gex

, accounts for the additional solution energy for what is historically termed a

regular solution

Gex¼X

i¼1

X

j >1

XiXjEij ½9

whereX is the mole fraction and Eijthe interaction

energy between the componentsi and j The system

free energy is thus

G ¼ Gþ Gidþ Gex ½10

1.17.4.1 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

Gex¼ XiXj

X

k

Lk;ijðXi XjÞk ½11

where L is the coefficient of the expansion in k, which can also have a temperature dependence typically of the form a þ bT Thus, a regular solu-tion 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

Gex¼ XAXBEBW ½12 Here,XAXBrepresents a random mixture of A and

B components and is thus the probability that A–B

is a nearest-neighbor pair, and EBWis the Bragg– Williams model energetic parameter

In a relevant example, Kayeet al.16

have generated a solution model for the five-metal white phase noted above and more extensively discussed inChapter2.20, 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 2 also shows a low-temperature miscibility gap, that is, two coexisting identically structured phases rich in either end member The excess Gibbs free energy expression for

Liquid

1669

1210 1183

Pd (fcc) + Rh (fcc)

XRh= 0.55

Pd

XRh= 0.33

Solid (fcc)

2236

1827

0.0 1000 1500 2000 2500

1000 1500 2000 2500

0.2 0.4 0.6 0.8 1.0

Model of Gürler et al.

Model of Jacob et al.

Figure 2 Computed Pd–Rh phase diagram with indicated data of Kaye et al.16illustrating 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.

the fcc phase was determined from an optimization

using tabulated thermochemical information together

with the phase equilibria and yielded

Gex¼ XPdXRh½21247 þ 2199XRh

 ð2:74  0:56XRhÞT ½13

1.17.4.2 Variable Stoichiometry/Associate

Species Models

As noted inChapter2.01, The Actinides Elements:

Properties and Characteristics; Chapter 2.02,

Thermodynamic and Thermophysical Properties

of the Actinide Oxides; and Chapter2.20, Fission

Product Chemistry in Oxide Fuels, modeling of

complex systems such as U–Pu–Zr and (U,Pu)O2 x

has been exceptionally difficult For example,

actinide oxide fuel is understood to be

nonstoi-chiometric almost exclusively due to oxygen site

vacancies and interstitials As a result, the

fluorite-structure 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

stoichi-ometry species that are mixed as subregular solutions

to fit experimental information.17–20This 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 [mO2¼ RT ln(pO2*)]

where pO* is a dimensionless quantity defined by2

the oxygen pressure divided by the standard state

pressure of 1 bar The UO2x 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 3illustrates 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

stoichio-metries for the constituent species yield differing

curves of ln(pO2*) versusf(x), with the most

appropri-ate matching the slope of 2 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

ade-quately represent the entire compositional range for

UO2x These are

UO2þx(high hyperstoichiometry, i.e., large values

ofx): UO2þ U3O7,

UO2þx (low hyperstoichiometry, i.e., smaller values ofx): UO2þ U2O4.5, and

UO2x(hypostoichiometric): UO2þ U1/3 The results of the models for UO2xare plotted in

Figure 4 together with the entire data set used for optimizing the system

The above models for UO2 x have been widely adopted, as has been a similar model of PuO2 x.16 These have also been combined to construct a suc-cessful model for (U,Pu)O2 x.16Lewiset al.22

used an analogous technique for UO2 x Lindemer23 and Runevallet al.24

have generated successful models of CeO2x Runevall et al.24

also used the method for NpO2x, AmO2x (with the work of Thiriet and Konings25), (U,Am)O2x, (Th,U)O2x, (U,Ce)O2x, (Pu,Am)O2x, and (U,Pu,Am)O2x They noted that results for the (Th,U)O2þxwere less successful per-haps because of the difficulty in the measurements

-4 -8

-16 -12

-20

-24

4

-4

0 Slope -2

Raw data Hagemark and Broli, 1673 K Roberts and Walter, 1695 K

f (X)

8

-8

Figure 3 The ln(pO 2 * ) dependence as a function of x for

UO2þxand of f(x) for several solid-solution species’ stoichiometries for an illustrative oxygen pressure– temperature–composition data set Coincidence with the theoretical slope of 2 indicates the proper solution model Reproduced from Lindemer, T B.; Besmann, T M.

J Nucl Mater 1985, 130, 473–488.

made near stoichiometry Osaka et al.26–28

used the approach to successfully represent the (U,Am)O2x,

(U,Pu,Am)O2x, and (Am,Th)O2xphases

1.17.4.3 Compound Energy Formalism

Regular or subregular solution and variable

stoichi-ometry representations, while relatively successful,

lack a sense of reproducing physical processes

Specifically, they are constrained with regard to

accu-rately 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 repre-sented by a formula, for example, (A,B)k(D,E)lwhere

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

0

-200

-400

-600

-800 500

(UO

2 ) exact

x in (UO

2 - )x

x in (UO

2+x

)

10

- 6

10 -6

10 -5

10 -4

10 -3

0.003

0.003

0.03

0.2 0.1

0.006 0.01

U–O liquid region

U–O liquid region

X = 0.3

0.006 0.01 0.03

0.1

0.15

0.2

0.25

0.27

10

-3

10

- 5

10

- 4

1000 1500 2000 2500

Temperature (K)

–1 )

3000

Figure 4 Oxygen potential plotted versus x for the models of UO2xof Lindemer and Besmann 17 overlaid with the entire data set used for the optimization.

of the standard Gibbs free energies in this case is the

sum of the values for the paired sublattice

constitu-ents, 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

sim-ple geometric additions with any necessary additional

configurational entropy contributions The entropy

contribution from mixing on the sublattice sites is

defined as

Gid¼ RTXzsys lnðys

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

exam-ple, AkDl: BkDl Again, a Redlich–Kister–Muggianu

formulation that includes expansion terms for

inter-actions between the constituents can be used:

Gxs ¼X

i

X

j

X

k

y1

iy2

jy1

kLi; j :k

þX

i

X

j

X

k

y2

ky1

iy1

jLk:i; j ½15

where the sums are associated with components on

each sublattice 1 and 2 and theL values are terms for

the interaction energies between cations i and j on

one sublattice when the other sublattice is occupied

only by cationk, and vice versa for the second term

The PuO2 xphase has been successfully

repre-sented by a CEF approach by Gueneauet al.30

The phase can be described by two sublattices with

vacan-cies only on the anion sites

(Pu4þ,Pu3þ)1(O2,Va)2

Including the end members, the constituent species

are then

(Pu4þ)1(O2)2, (Pu4þ)1(Va)2, (Pu3þ)1(O2)2, and (Pu3þ)1(Va)2

A schematic of the relationship between the constituents is seen in Figure 5 where the corners represent each of the constituents listed above The charged constituents must sum to neutrality, and the line designating neutrality is seen inFigure 5 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 inFigure 6where oxygen potential isotherms overlay the phase diagram and which shows

mO2results of models for other phases in the system The CEF approach has recently begun to be more widely applied to nuclear fuels Besides the PuO2x system noted above, Gueneauet al.31

also applied the model to accurately describe solid solution phases

in the U–O system, as has Chevalieret al.32

who also addressed the U–O–Zr ternary system.33 Kinoshita

et al.34

used a sublattice approach to model fluorite-structure oxides including ThO2 x and NpO2 x, although they did not include charged ionic cations and anions on the sublattices Zinkevichet al.35

suc-cessfully modeled the CeO2 xphase 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 nonstoichio-metry is as a function of defects in the ideal lattice

Neutral line PuO1.5= (Pu 3+ )1(Va1/4, O 2 -3/4)2

(0)

(Pu 3+ )1(O 2 -)2

4+ )1(O 2 -)2= PuO2 (0)

(Pu 3+ )1(Va)2

( +4)

Figure 5 Compound energy formalism sublattice model illustration of the components and their charge in a

two-dimensional representation after Gueneau et al.31

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

(seeSection 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,36Stanet al.,37

and Nerikaret 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 Uþ3, Uþ4, Uþ5, and Uþ6

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

O

O ¼ Oi00þ V

O

‰O2þ 2UU ¼ O00

i þ 2U U

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 hyperstoi-chiometric urania is plotted as a function of defect concentrationxa The relationships were also used to compute oxygen potentials as a function of stoichi-ometry 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 accu-rate representations of the complex liquids

0

-4 -8 -12 -16 -20 -24 -28 -32 -36 1.5 1.6

O/Pu ratio 1.7 1.8 1.9 2.0

O2

Figure 6 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 PuO2xphase Reproduced from Gueneau, C.; Chatillon, C.; Sundman, B J Nucl Mater 2008, 378, 257–272.

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

mis-cibility 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,

G¼ XAGAþ XBGBþ XA 2 BGAB ½16

Consequently, ideal mixing among end members and associates generates the entropy contribution

Gid¼ RTðXAlnXAþ XBlnXBþ XA 2 BlnXA 2 BÞ ½17 Should a nonideal term providing positive interac-tion energies be needed to properly address a misci-bility 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

Gex¼ XAXA 2 B

X

k

Lk;A:A2 BðXA XA 2 BÞk ½18 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 crys-talline 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 gener-ated from crystalline phases of the same stoichiome-try 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 8 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

0.25

0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00 -12 -10 -8 -6

0.2

0.15

0.1

0.1

x in UO 2+x

Xa

0.15 0

0.05

1273 K

1273 K

Model Model

Nakamura and Fujino

UO2+x

log10 pO2 (atm)

Oi

Oi

1373 K

1373 K

0.05

VO• •

VO• •

UU•

Figure 7 (a) Concentrations of defect species in UO2þxrelative to the concentration of oxygen sites in the perfect lattice, as a function of nonstoichiometry, calculated with a defect model (b) UO2þxnonstoichiometry as a function of partial pressure of oxygen (Dashed line is model-derived and solid line are results of Nakamura and Fujino36and

Stan et al.37)