Transcript From Quantum to Continuum Mechanics with Thermodynamics

On the modelling of the phases in O-U-Zr
Bo Sundman
MSE, KTH, Sweden
Christine Gueneau, Delphine Labroche,
CEA, Saclay, France
Christian Chatillon, Mehdi Baichi,
LTPCM, Grenoble, France
Thermodynamic Modelling
• Thermodynamics provide relations between many different
materials properties like amount of phases and their
compositions, heat of transformation, partial pressures etc.
• Experimental thermodynamics has proved its value for
materials research for more than 100 years.
• Modern multicomponent materials would require very
extensive experimental work in order to establish the
relevant data for phase equilibria and thermodynamic
properties.
• Modelling is the only possible technique to reduce
experimental work and to increase the value of each new
experiment by allowing accurate extrapolation.
• Thermodynamics provide the most important information
for a phase transformation: the final equilibrium state
Hierarchically Structured Materials Design System
Hierarchy of computational design
models and experimental tools
LM, TEM
MQD, DSC
LM, TEM
JIC, i
SAMNS, XRD
APFIM, AEM
 , H
Transform.
Design
Micromech.
Design
Nano
Design
TC/MART
CASIS
ABAQUS/SPO
TC, V
TC/DICTRA
ABAQUS/EFG
SAM
KGB()
Quantum
Design
FLAPW
DVM
Thermo-Calc/DICTRA
Developed by Northwestern Univ, 1997
All kinds of modelling and simulations depend heavily on
the availability of cheap and fast computing power. A lot
has happend since this picture from 1970 …
The CALPHAD technique uses thermodynamic models
based on
• experimental information on phase diagrams
(solubilties, invariant temperatures etc) and
• experimental data on thermochemistry (enthalpies,
chemical potentials etc) and
• theoretical calculations like ab inito and
• empirical rules using
• physically realistic models with adjustable parameters
describing the Gibbs energy of each phase in the system.
Calculated binary and
ternary phase diagram
for the system
Al-Cu-Mg-Zn
Using the CALPHAD technique thermodynamic
assessments of more than 1000 binary, ternary and
higher order systems have been made.
This work can only be successful by joint
international collaboration and requires international
agreement of a number of important critical
quantities:
• the thermodynamic model to use for the different
phases, and
• the properties of elements in different metastable
states (lattice stabilities), for example the
properties of Zr in an FCC crystal lattice
With 30 years experience developing CALPHAD
databases it has been clearly shown that with good
models it is possible to calculate accurately the stable
state of alloys with 8-12 components using carefully
made assessments of the most important binary and
ternary subsystems.
It is interesting to compare CALPHAD to a similar
technique developed by materials physicists in close
connection to ab initio calculation of materials
properties. In these techniques very complex
thermodynamic models are used to calculate the
equilibrium of a system, for example Cluster Variation
Method (CVM) or Monte Carlo (MC). The main
differences with CALPHAD are:
• Systems calculated separately using CVM/MC
techniques cannot be combined and extrapolated
to higher order systems.
• In most cases the CVM/MC calculations use only
ab inito data and the calculated results are far
away from experimental data.
• One important feature with the ab initio calculations is
that they can determine properties of a phase that is
not stable, i.e. when it is not possible to measure its
properties. But whenever possible calculations should
be checked by experiments.
• The quantities from ab initio calculations can be
combined with experimental data in a CALPHAD
assessment to give an accurate and reliable
thermodynamic description, rather than just compared
with experiments as with usual CVM or MC
calculations.
In CALPHAD the Gibbs energy for each phase is
modelled separately and rather simple models are used
The gas phase is normally assumed to be an ideal
mixing of the gas species i with constituent
fraction yi:
Gm = Si yi oGi + RT Si yi ln( yi )
The liquid phase can be modelled in many different
ways depending on the type of components. For
metallic liquids one usually has a substitutional
regular solution model with the mole fractions xi:
Gm = Si xi oGi + RT Si xi ln( xi ) + EGm
The excess Gibbs energy, EGm, takes interaction
between the different constituents into account.
A simple series in fraction is used:
EG = S S S x x (x – x )n L
m
i j>i n i j
i
j
ijn + ternary…
More complicated models including associates, ions,
quasichemical configurational entropy etc are used to
describe molten salts, oxides and other liquids with a
strong tendency for short range order.
An important feature is that different models should
be compatible, for example it should be allowed in
the model for a liquid that it gradually changes from
metallic to ionic with temperature or composition.
The ionic liquid model has been developed to handle both
metallic and ionic liquids. The Gibbs energy expression is
Gm = Si Sj yi yjoGij + Q yVa Si yioGi + Q Sk ykoGk +
RT(PSi yiln(yi) + Q(Sj yjln(yj)+yValn(yVa)+Sk ykln(yk))) +
EG
m
Where i denotes cations like Fe+2, U+4 etc, j denotes anions
like O-2, Cl-1 and k denotes constituents that are neutral in
the liquid like C. The factors P and Q depend on
composition to ensure that the liquid is always electrically
neutral.
This model has successfully been applied to liquid in
systems like Fe-Ca-Si-O as well as the U-O.
Solid phases are usually crystalline and the crystal
structure must be taken into account in the modelling.
This can be done using the sublattice model. This
can handle interstitials like carbon in steels, oxides
like spinels and defects in intermetallic phases like
Laves, , L12 etc with 2 to 8 sublattices. The Gibbs
energy for a two-sublattice phase can be written:
Gm=Si Sj y’i y”j oGm + RT Ss as Si y(s)i ln( y(s)i) + EGm
where y’i and y”j are the site fractions on the two
sublattices and as the number of sites on sublattice s.
oG is the Gibbs energy of formation of the
ij
compound ”ij” and EGm the excess Gibbs energy.
Sublattice model for UO2
(+2)
4VO
(U+3,U+4,U+6)1(O-2,VA)2(O-2,VA)1
=
(3,4,6)1(O,V)2(O,V)1
(-2)
4OO
(0)
4OV
(+1)
3VO
(+4)
6VO
(+4)
4VV
(-3)
3OO
(-1)
3OV
(+3)
3VV
(0)
6OO
(+2)
6OV
(+6)
6VV
Binary U-O
THERMO-CALC (2004.09.03:11.51) :
DATABASE:USER
P=100000, N=1;
4500
liquid
4000
TEMPERATURE_KELVIN
The calculated U-O
system using the
ionic liquid model
and a complex
defect model for the
UO2 phase.
There is a wide
miscibility gap in
the liquid phase.
gas
3500
3000
2500
2000
1500
1000
500
0
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
MOLE_FRACTION O
U-O with experiments
THERMO-CALC (2004.09.03:12.13) :
DATABASE:USER
P=100000, N=1;
3200
The same calculated
diagram compared with
experimental data
TEMPERATURE_KELVIN
3000
2 liquids
2800
2600
2400
2200
liquid + UO2
2000
1800
1600
1400
1200
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
MOLE_FRACTION O
UO2 withTHERMO-CALC
experiments
(2004.09.03:12.15) :
DATABASE:USER
P=100000, N=1;
3300
The calculated
solubility range of
UO2 with
experimental data
TEMPERATURE_KELVIN
3000
2700
2400
2100
1800
1500
1200
900
600
300
0.60
0.65
0.70
MOLE_FRACTION O
0.75
Enthalpy
in
UO
2
THERMO-CALC (2004.09.03:12.23)
:
DATABASE:USER
P=100000, N(O)=2, N(U)=1;
100
90
80
70
Enthalpy J/mol
Calculated and
experimental values
of the heat content
of UO2. This is
calculatated from
the same model as
the phase diagram.
60
50
40
30
20
10
0
0
500
1000 1500 2000 2500 3000 3500
T
PO2 in UO2
THERMO-CALC (2004.09.03:12.32) :
DATABASE:USER
T=2400, P=100000, N=1.;
1700 K
2000
-6
1600 K
2100
1500 K
2250
1400 K
2300
10
1300 K
2400
10-12
1200 K
-14
1100 K
10
-8
10
Partial pressue O2
Calculated and
experimental
partial
pressures of O2
in the UO2
phase.
10-4
-10
10
1000 K
-16
10
900 K
-18
10
10-20
1.90
800 K
1.95
2.00
2.05
N(O)/N(U)
2.10
2.15
2.20
Metstable extrapolations
The CALPHAD models provide values of the Gibbs
energy at temperatues and compositions outside the
stable range of the phase and this is one of the key
features of CALPHAD.
In the 1980-ies there were fierce discussions between
Calphadists and chemists if the Gibbs energy
function outside the stable range of the phase was
meaningful but eventually it was accepted.
Still today it seems that some chemists and
physicists doubt that the Gibbs energies calculated
from a models for a phase are meaningful outside the
stable range of the phase.
Congruent melting for UO2
THERMO-CALC (2004.09.03:12.47) :
DATABASE:USER
N(O)=2, N(U)=1,FIXED PHASES: GAS=0;
10
87Bob
1
P (MPa)
A more unusual kind
of experimental data,
the total pressure at
the congruent boiling
temperature at various
tempertures. The
calculated line fits the
experimetal data well.
.1
.01
2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0
10000/T
Binary Zr-O
THERMO-CALC (2004.09.03:12.35) :
DATABASE:USER
P=100000, N=1;
3000
TEMPERATURE_KELVIN
The assessed phase
diagram for the Zr-O
system. The high
temperature ZrO2
phase is the same
structure type, C1, as
the UO2 phase.
There is no
miscibility gap in the
liquid phase.
3300
liquid
2700
2400
2100
1800
1500
1200
HCP
900
600
300
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
MOLE_FRACTION O
A
A
A
A
A
A
A
A
A
H
R
G
A
G
D
C
A
Binary U-Zr
THERMO-CALC (2004.09.03:12.37) :
DATABASE:USER
P=100000, N=1;
2200
TEMPERATURE_KELVIN
In the metallic UZr system there is
complete
solubility in the
liquid and the
BCC phases.
2000
liquid
1800
1600
1400
BCC
1200
1000
800
600
400
200
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
MOLE_FRACTION ZR
THERMO-CALC (2004.09.03:16.06) :
DATABASE:USER
T=1273, P=100000, N=1.;
O-U-Zr at 1273 K
1.0
0.9
O
0.8
_F
RA
CT
IO
N
0.7
0.6
0.5
MO
LE
An isothermal section
at 1273 K for the
ternary O-U-Zr
system. There is little
or no ternary
solubilities.
0.4
0.3
0.2
BCC+HCP
+UO2
0.1
0
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
MOLE_FRACTION ZR
THERMO-CALC (2004.09.03:14.16) :
DATABASE:USER
T=2273, P=100000, N=1;
O-U-Zr at 2273 K
1.0
0.9
O
0.8
_F
RA
CT
IO
N
0.7
0.6
0.5
MO
LE
At 2273 K the liquid
phase extends far into
the system on the Zr-O
side.
The C1 phase extends
across the system but
ZrO2_tetr is still stable.
There is a miscibility
gap in the C1 phase
close to UO2
0.4
0.3
0.2
0.1
liquid
0
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
MOLE_FRACTION ZR
O-U-Zr at 2773 K
At 2773 K the liquid
phase forms a
closed miscibilty
gap inside the
ternary.
THERMO-CALC (2004.09.03:20.56) :
DATABASE:USER
T=2773, P=100000, N=1;
Top part of O-U-Zr at 2773 K
1.00
0.95
NO
0.90
MO
LE
_F
RA
CT
IO
This is the high
oxygen part of the
same isothermal
section. There is a
liquid phase on both
sides of the C1
phase and still a
miscibility gap in
the C1 phase for
small Zr additions.
0.85
0.80
gas+liquid
0.75
0.70
0.65
liquid+C1
0.60
0
0.05
0.10
0.15
0.20
0.25
MOLE_FRACTION ZR
0.30
0.35
0.40
O-U-Zr at 2973 K
THERMO-CALC (2004.09.03:16.08) :
DATABASE:USER
T=2973, P=100000, N=1;
1.0
0.9
_F
RA
CT
IO
N
0.7
0.6
0.5
MO
LE
At 2973 K the C1 phase
is no longer stable on
the Zr-O side.
O
0.8
0.4
0.3
0.2
0.1
0
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
MOLE_FRACTION ZR
Section UO2-ZrO2
THERMO-CALC (2004.09.03:13.36) :
DATABASE:USER
N=1, P=100000,X(U)+X(ZR)=3.33333333E-1;
3500
liquid
3000
TEMPERATURE_KELVIN
A calculated
section from UO2
to ZrO2 together
with experimental
data showing the
liquid, the C1
phase and the low
temperature ZrO2
phases.
Baichi 2000
Paschoal 1987
Une 1983
Smith 1981
Romberger 1967
Baes 1963
Cohen 1963
Voronov 1958
Lambertson 1953b
C1
2500
T
2000
C1+T
1500
C1+M
1000
500
0
0.05
0.10
0.15
0.20
X(*,ZR)
0.25
0.30
Development of CALPHAD databases
The existing databases for steels, superalloys,
aluminium, ceramics etc. can calculate thermodynamic
properties for commercial alloys with up to 12
components and are used by industry.
Each such database represent typically 50-100 manyears
of assessment work by scientists and graduate students.
More important, each database contain several 1000
manyears of experimental work and can save much more
experimental work in the future as calculations make it
possible to select critical experiments.
Conclusions
• Thermodynamic modelling of alloys and oxide
systems provide a good estimate of the
thermodynamic data for multicomponent systems.
• Models for liquids and solid with defects is
complex and an important field of research in
modelling, experimentally and ab initio.
• Thermodynamic data gives information on phase
transformations from metastable states as they
provide information on the stable state the system
tries to reach.
End of lecture
That’s all