On the modelling of the phases in O-U-Zr

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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
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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

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Hierarchically Structured Materials Design System
Hierarchy of computational design models and
experimental tools
LM, TEM MQD, DSC
TC/MART CASIS
LM, TEM JIC, ?i
ABAQUS/SPO TC, ?V
SAMNS, XRD APFIM, AEM ??, H
TC/DICTRA ABAQUS/EFG
Connection with TQ/API
SAM KGB(??)
FLAPW DVM
Thermo-Calc/DICTRA
Developed by Northwestern Univ, 1997
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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
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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.

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Calculated binary and ternary phase diagram for
the systemAl-Cu-Mg-Zn
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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

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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.
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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.

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• 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.

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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
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The excess Gibbs energy, EGm, takes interaction
between the different constituents into account.
A simple series in fraction is used EGm Si
Sjgti Sn xi xj (xi xj)n Lijn 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.
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• 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)))
• EGm
• Where i denotes cations like Fe2, U4 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.

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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, s,
L12 etc with 2 to 8 sublattices. The Gibbs
energy for a two-sublattice phase can be
written GmSi Sj yi yj oGm RT Ss as Si
y(s)i ln( y(s)i) EGm where yi and yj are the
site fractions on the two sublattices and as the
number of sites on sublattice s. oGij is the
Gibbs energy of formation of the compound ij
and EGm the excess Gibbs energy.
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Sublattice model for UO2
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Binary U-O
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
liquid
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U-O with experiments
2 liquids
The same calculated diagram compared with
experimental data
liquid UO2
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UO2 with experiments
The calculated solubility range of UO2 with
experimental data
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Enthalpy in UO2
Calculated and experimental values of the heat
content of UO2. This is calculatated from the
same model as the phase diagram.
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PO2 in UO2
Calculated and experimental partial pressures of
O2 in the UO2 phase.
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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.
  

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Congruent melting for UO2
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.
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Binary Zr-O
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.
liquid
HCP
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Binary U-Zr
In the metallic U-Zr system there is complete
solubility in the liquid and the BCC phases.
liquid
BCC
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O-U-Zr at 1273 K
An isothermal section at 1273 K for the ternary
O-U-Zr system. There is little or no ternary
solubilities.
BCCHCP UO2
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O-U-Zr at 2273 K
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
liquid
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O-U-Zr at 2773 K
At 2773 K the liquid phase forms a closed
miscibilty gap inside the ternary.
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Top part of O-U-Zr at 2773 K
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.
gasliquid
liquidC1
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O-U-Zr at 2973 K
At 2973 K the C1 phase is no longer stable on the
Zr-O side.
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Section UO2-ZrO2
A calculated section from UO2 to ZrO2 together
with experimental data showing the liquid, the C1
phase and the low temperature ZrO2 phases.
liquid
C1
T
C1T
C1M
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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.
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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.

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End of lecture
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Thats all