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
aþbT
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 ¼ E þ PV À TS

½1Š

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 ¼ H À TS

½2Š



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 þ

Cp dT

½3Š

Cp =T dT

½4Š

298

and to entropy by
ðT
S ¼ S298 þ
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 ¼ A ¼ BT þ CT lnT þ DT 2 þ ET 3 þ F =T


½5Š

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 0. 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 ¼ ð@G=@ni ÞT ;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 1.
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 UO2Æx 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 1 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 ¼ RT
ni ln ni
½8Š


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 þ 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 Þk
½11Š
G ex ¼ 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 2 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 ¼ G þ G id þ G ex

459

0.4

0.6

0.8

XRh

1.0
Rh

16

Figure 2 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)O2Æx
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 1 bar. The UO2Æx 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 3 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 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 adequately represent the entire compositional range for
UO2Æx. These are

-12
In (pO2* )

G ex ¼ XPd XRh ½21247 þ 2199XRh
À ð2:74 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

0
f (X)

4

8

Figure 3 The ln(pO2* ) dependence as a function of x for
UO2þx 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 2 indicates the proper solution
model. Reproduced from Lindemer, T. B.; Besmann, T. M.
J. Nucl. Mater. 1985, 130, 473–488.

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

of x): UO2 þ U3O7,
UO2þx (low hyperstoichiometry, i.e., smaller
values of x): UO2 þ U2O4.5, and
UO2Àx (hypostoichiometric): UO2 þ U1/3.
The results of the models for UO2Æx are plotted in
Figure 4 together with the entire data set used for
optimizing the system.
The above models for UO2Æx 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)O2Æx .16 Lewis et al.22 used an
analogous technique for UO2Æx . 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)O2Æx, (Th,U)O2Æx, (U,Ce)O2Àx,
(Pu,Am)O2Æx, and (U,Pu,Am)O2Æx. They noted that
results for the (Th,U)O2þx were less successful perhaps because of the difficulty in the measurements


Computational Thermodynamics: Application to Nuclear Materials

)

UO

0

(

n


xi

x
2+

7

0.2

25

0.

0.2

461

U–O
liquid
region

0.15
0.1

0.03
0.01
0.006

-200


0.0 0.0
06
1

03
0.0

10-3

-400

0.0

10 -4

3

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

6

0.2

10 -

-600


10 -4

0.1

5

-800
500

1000

1500

2000

2500

3000

Temperature (K)
Figure 4 Oxygen potential plotted versus x for the models of UO2Æx 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 lnðyjs Þ
½14Š
G id ¼ 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 1 and 2 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
(Pu4þ,Pu3þ)1(O2À,Va)2.
Including the end members, the constituent species
are then

(Pu4þ)1(O2À)2,
(Pu4þ)1(Va)2,
(Pu3þ)1(O2À)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 in Figure 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 in Figure 6 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 5 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

0
-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 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 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 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
00

OOÂ ¼ Oi þ VOÁÁ

00

‰O2 þ 2U UÂ ¼ Oi þ 2UUÁ
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

0

Nakamura and
Fujino

0.08

0.04
1273 K

UO2+x

(b)

-10
-8
log10 pO2 (atm)

-6

Figure 7 (a) Concentrations of defect species in UO2þx relative to the concentration of oxygen sites in the perfect
lattice, as a function of nonstoichiometry, calculated with a defect model. (b) UO2þx 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 þ XB GB þ XA2 B GA2 B

½16Š

Consequently, ideal mixing among end members
and associates generates the entropy contribution
G id ¼ RT ðXA ln XA þ XB ln XB þ XA2 B ln XA2 B Þ 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 ðXA À XA2 B Þk
½18Š
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 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


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 8 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 9.
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 do 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 do 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 9 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.
ðA À AÞ þ ðB À BÞ ¼ 2ðA À BÞ

½19Š

A parameter, Z, represents the nearest-neighbor
coordination number such that each component
forms Z pairs. For one mole of solution,
ZXA ¼ 2nAA þ 2nAB


½20Š

ZXB ¼ 2nBB þ 2nAB

½21Š

where the moles of each pair are nAA, nBB, and nAB.
The relative proportions of each pair is Xij, where
Xij ¼ nij =ðnAA þ nBB þ nAB Þ

½22Š

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 ¼ ðXAB =2ÞEQM

½23Š

where EQM is the quasichemical model energetic
parameter, and a configurational entropy

Sconfig ¼ À RðXA ln XA þ XB ln XB Þ
À RðZ=2ÞðXAA lnðXAA =XA2 Þ þ XBB lnðXBB =XB2 Þ
þ XAB lnðXAB =2XA XB ÞÞ

½24Š

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 ¼ ðEQM =2ÞXAB þ 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
1 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 1 atm and of 1 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 UO2Æx
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 comments.
Research supported by the U.S. Department of
Energy, Office of Nuclear Energy, under the Fuel
Cycle Research and Development and Nuclear
Energy Advanced Modeling and Simulation Programs. This manuscript has been authored by
UT-Battelle, LLC, under Contract No. DE-AC0500OR22725 with the U.S. Department of Energy.
The U.S. Government retains and the publisher, by
accepting the article for publication, acknowledges
that the U.S. Government retains a non-exclusive,
paid-up, irrevocable, world-wide license to publish
or reproduce the published form of this manuscript, or allow others to do so, for U.S. Government
purposes.

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23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
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