EPJ Web of Conferences 14, 01003 (2011)
DOI: 10.1051/epjconf/20111401003
© Owned by the authors, published by EDP Sciences, 2011
“ Fundamentals of Thermodynamic Modelling
of Materials ”
November 15-19, 2010
INSTN – CEA Saclay, France
PROFESSOR & TOPIC
Bo SUNDMAN
INSTN-CEA Saclay and KTH, Sweden
Thermodynamic
Modelling
[01003]
Organized by
Bo SUNDMAN [email protected]
Constantin MEIS [email protected]
This is an Open Access article distributed under the terms of the Creative Commons Attribution-Noncommercial License 3.0, which
permits unrestricted use, distribution, and reproduction in any noncommercial medium, provided the original work is properly cited.
Article available at http://www.epj-conferences.org or http://dx.doi.org/10.1051/epjconf/20111401003
Thermodynamic Modelling
Bo Sundman
INSTN/CEA Saclay winterschool 2010
The secrets of multicomponent modelling of materials
Content
• Lecture: Equation of State, Gibbs energy. Regular
solution, sublattice model, lattice stabilities, enthalpy
models, entropy models, associate models,
quasichemical models, CVM, configurational terms,
defect models.
• Software session 1: Calculation of phase equilibria,
phase diagrams, chemical potentials, driving force,
thermodynamic factor, phase transformations
• Software session 2: Assessment of Fe-Ti system
01003-p.2
Thermodynamic models
• A model for a phase may contain real and
fictitious species. These species, called the
constituents, contribute to the entropy of mixing.
• The ideal entropy of mixing of a phase having the
components as constituents comes from Bolzmann
S=R ln( (Ni)! / (Ni)! )
• This can be derived in two ways, either
distributing different atoms on a given set of
lattice points or from the statistical mechanics of
an ideal gas.
Thermodynamic models
• From Stirlings formula per mole of phase
Sm= -R i xi ln(xi)
• Mole fraction of components, xi = Ni/i
• Constituent fraction, yi, is equal to the
amount of the constituent divided by the
total amount of constituents on a lattice. For
a gas phase each molecule has a constituent
fraction.
01003-p.3
Thermodynamic models
• Elements – those from the periodic chart
• Species – an element or a combination of
elements that forms an entity, like H2O, CO2,
Fe+2
• Constituents are the species that exist in a
phase. A constituent can be real or fictitious.
• Components is an irreducible subset of the
species
Thermodynamic models
• Each phase is modelled separately
• Phases with no compositional variation has
just an expression G(T,p). That is very
simple to handle at low pressures.
• Phases with a small compositional variation
can be very difficult to model as one should
take into account the different types of
defects that cause the non-stoichiometry
01003-p.4
Thermodynamic models
• The temperature dependence of a Gibbs
energy parameter is normally a polynomial
in T, including a TlnT term from the heat
capacity
• G = a + bT + cTlnT + dT2 + …
• Note that the enthalpy, entropy, heat capacity
etc can be calculated from this G.
• The pressure dependence, except for a
pressure independent volume, is more
complicated and will be discussed later
Thermodynamic model
• Properties at low temperature, normally below
300K, is normally not modelled.
• The Gibbs energy at low temperature has a
complicated T-dependence (Debye model) that is
not easy to combine with the higher temperature
properties.
• Enthalpy data at 0 K, from ab initio calculations,
are useful for fitting high temperature data
01003-p.5
Thermodynamic models
• Phases with extensive compositional
variation are the gas and liquid or have
usually rather simple lattices for example
fcc (A1), bcc (A2) and hcp (A3).
• Some more complex lattices belong to
families of simpler lattices, like B1 is A1
with interstitials, B2 is ordered A2 etc.
That should be taken into account in the
modelling
Crystallographic data
http://cst-www.nrl.navy.mil/lattice/
A1
B1
L10
A2
B2
D03
01003-p.6
L12
L21
Thermodynamic models
• From the thermodynamic models one can
calculate various thermodynamic properties of a
system, like heat of transformation, chemical
potentials, heat capacities etc
• One may also calculate the phase diagram or
metastable extrapolations of the phase diagram
• One may make more reliable extrapolations
in temperature and composition than if one
extrapolated a single property
• They can be used in software for simulations
of phase transformations
Thermodynamic models
• The Gibbs energy per mole for a solution phase is
normally divided into four parts
• Gm = srfGm – T cfgSm + EGm + physGm
•
•
•
•
srfG
m is the surface of reference for Gibbs energy
cfgS
m is the configurational entropy
EG is the excess Gibbs energy
m
physG is a physical contribution (magnetic)
m
01003-p.7
Physical property data
The only explicit physical model used at present is one for the
magnetic contribution to the heat capacity as a function of the Curie
temperature and the Bohr magneton number proposed by G Inden.
Gmagn = f() ln( +1 )
Where f(t) is a function fitted the the experimental data for
The magnetic contribution to the heat capacity for Fe, Ni and Co,
and is T/Tc where Tc is the Curie temperature and is the Bohr
magneton number.
Both Tc and can depend on composition. They may also depend
on the pressure.
Physical property data
The contribution due to the ferromagnetic transition in some
elements like Fe is shown below. The left fgure is the effect on the
heat capacity of pure Fe and the right figure show how the Curie
temperature depend on the composition in Fe-Cr system.
1200
60
55
1000
800
45
TC(BCC)
FUNCTION CP
50
40
35
600
400
Non-magnetic Cp
30
200
25
0
20
0
500
1000
1500
2000
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
X(CR)
T
01003-p.8
Thermodynamic models
Modelling by physicists have mainly
concentrated on finding a good
configurational entropy (Quasichemical,
CVM, Monte Carlo) to describe main
features or a specific detail.
Modelling among material scientists has
mainly concentrated on finding a good
excess Gibbs energy to reproduce the
experimental data.
Regular solutions
• Regular solution models are based on ideal
entropy of mixing of the constituents. In the
general case these are different from the
components and their fraction is denoted yi
• Gm = i yi oGi + RT i yi ln(yi) + EGm
• oGi is the Gibbs energy of pure constituent i
in phase • EGm is the excess Gibbs energy
01003-p.9
Excess energies
• If A and B atoms occupy neighbouring
lattice sites the energy of the AB bond is
related to that of an AA and a BB bond by
• EAB = AB - 0.5 (AA + BB)
• If this energy is negative the atoms like to
surround themselves with the other kind of
atom.
• If the energy is positive there is a demixing
Regular solution model
•
•
•
•
The excess Gibbs energy for a binary system
EG = y y L
m
i
j>i i j ij
Lij = (yi – yj) Lij
(Redlich-Kister)
Other types of polynomial are possible but all
are identical in the binary case. However,
they will differ in ternary extrapolations and
thus the most symmetrical is preferred
• Lij = 0.5 z Eij where z is the number of nearest
neighbours.
01003-p.10
TEMPERATURE_KELVIN
900
800
700
600
500
400
300
200
0
0.2
0.4
0.6
0.8
1.0
MOLE_FRACTION Y
Ideal liquid interaction,
solid interaction 0,
+10000 and –10000
Redlich-Kister coefficients
The contribution to
the excess enthalpy as
a function of
composition for the
first three coefficients
of the RK series, all
with the same value,
10000 J/mol.
01003-p.11
2003-03-09 20:20:47.58 output by user bosse from GIBBS
1000
Ternary regular solution parameter
• EGm = yi yj yk L ijk
• L ijk = vi 0Lijk + vj 1Lijk + vk 2Lijk
• vi = yi + (1 – yi – yj – yk)/3
In the ternary system vi = yi. In higher order
systems vi = 1 always which guarantes the
symmetry.
Lattice stabilities
A solution model for a phase often extend from
one pure component to another even if one, or
both, of them may not exist as stable in that
phase.
These ”lattice stabilities” of the metastable states
of elements was first introduced by Larry
Kaufman and must be agreed internationally to
make assessments compatible. Most commonly
used are those by SGTE, published in Calphad
by Dinsdale 1991
01003-p.12
Liquidus extrapolations for Cr
FCC
FCC
Lattice stability for Cr
First principle calculations have shown fcc-Cr
is mechanically unstable, thus it is impossible
to calculate the energy difference between fcc
and bcc for pure Cr.
It has been accepted that the ”Calphad” value
is reasonable within the range Cr dissolves in
a stable fcc phase and as long as one does not
believe it represents a real fcc phase.
01003-p.13
Dilute solution model
• Based on Henry’s law for the activity of the solute
and Raoult’s law for the activity of the solvent.
”Epsilon” parameters describe the activity in more
concentrated solutions.
• The assumption that Raoults law is true for
multicomponent systems is wrong
• Dilute models are thermodynamically
inconsistent (they do not obey the GibbsDuhem equation) and cannot be used in
software for Gibbs energy minimizations.
Sublattice model
Crystalline phases with different types of
sublattices for the constituents can be
described with the sublattice model.
Different constituents may enter in the
different sublattices and one assumes ideal
entropy of mixing on each sublattice. The
simplest case is the reciprocal system
(A,B)a(C,D)c
01003-p.14
Sublattice model
• The Gibbs energy expression for (A,B)a(C,D)c
• srfGm = i j y’i y”j oGij
• cfgSm = -R(a i y’i ln(y’i) + c j y”j ln(y”j))
• oGij is the Gibbs energy of formation of the
compound iajc , also called “end members”.
• a and c are the site ratios
• The excess and physical contributions are as
for a regular solution on each sublattice.
Reciprocal square
(B:D)
0.9
LA,B:D
0.8
Constituent fraction D
The square represent
the constitutional square
for a sublattice model
(A,B)a(C,D)c
The corners represent
the 4 end members oGij
and the sides represent
the 4 interaction
parameters Lij:k and Li:jk
(A:D)
1.0
0.7
0.6
0.5
LA:C,D
LB:C,D
0.4
0.3
0.2
0.1
LA,B:C
0
0
(A:C)
01003-p.15
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
(B:C)
Constituent fraction B
Sublattice model
Excess Gibbs energy for (A,B)a(C,D)c
EG
m
= y’Ay’B(y”C L A,B:C + y”D L A,B:D) +
y”Cy”D(y’AL A:C,D + y”B L B:C,D) +
y’Ay’By”Cy”D L A,B:C,D
Each L can be a Redlich-Kister series
Sublattice model
• The sublattice model has been used extensively
to describe interstitial solutions, carbides,
oxides, intermetallic phases etc.
• It is often called the compound energy
formalism (CEF) as one of its features is the
assumption that the compound energies are
independent of composition and it includes
several models as special cases.
• Note that the Gm for sublattice phases is usually
expressed in moles for formula units, not moles
of atoms as vacancies may be constituents.
01003-p.16
Sublattice model
• Sublattices are used to describe long range
order (lro) when the atoms are regularly
arranged on sublattices over large distances.
• Short range order (sro) means that the
fraction of atoms in the neighbourhood of
an atom deviate from the overall
composition. There are special models for
that.
Connection with first principle calculations
From first principles one may calculate the energy at
0K for different configurations of atoms on specific
lattices. These energies can be expanded in different
ways to describe disordered states for compositions in
between the calculated configurations. A popular
model to use is the Cluster Expansion Method (CEM)
by Connally-Williams. The cluster energies can then
be used in a CVM or Monte Carlo (MC) calculation of
the phase diagram for example.
For phase diagram calculations of ordering in binary
systems with fcc lattices it is very important to include
the short range order but for carbides and intermetallic
phases like it is less important.
01003-p.17
Connections with first principle calculations
The energies from a first principle calculation can also be used
directly in a sublattice model if the configurations correspond
to the end members. For fcc there are theoretically 3 ordered
compounds, two with L12 and one with L10 structure. But
like in the Al-Ni system below some may be metastable and
their energies must be calculated using ab initio techniques.
Cluster Variation Method
• An improved method to treat short range order in
crystalline solids was developed 1951 by Kikuchi
and called Cluster Variation Method (CVM). It can
treat arbitarily large clusters of lattice sites but the
entropy expression must be derived for each lattice.
• Even for binary systems it can be rather
cumbersome to use CVM and for multi-component
systems it is impossible to apply. Anyway, for most
multicomponent phases the contribution to Gibbs
energy due to sro is small.
01003-p.18
Models for the liquid
• The liquid phase in metallic system is most
often modelled with a substitutional regular
solution model.
• For metal-nonmetal systems with strongly
assymetric properties like miscibility gaps
or rapid changes of activities the associated,
quasichemical or ionic liquid model my be
more appropriate.
Thermochemical properties for Fe-S
The phase diagram
and some thermodynamic properties:
activity, enthalpy and
entropy at 2000 K
01003-p.19
Associated solutions
• These are identical to regular solutions except that
one has added one or more fictitious constituents,
for example FeS in liquid Fe-S. The reason to
introduce this is to describe short range order around
the FeS composition.
• A parameter oGFeS describe the stability of the
associate.
• Interaction parameters between Fe-FeS and FeS-S
are added to those between Fe-S. It can thus be
modelled similarly to a ternary system.
• Note that a gas phase is similar to an associated
solution (without excess parameters) but in this case
the constituents are real.
Associated model
• The model assumes that one or more
”associates” are formed in the liquid. The
associate have all the properties of a real
species but cannot be verified by
experimental methods (if the species can be
verified experimentally then it is no longer
an asociated model but a reality)
01003-p.20
Associated model
• Gm = xA0GA+ xAB0GAB+ xB0GB+
RT(xAln(xA)+ xABln(xAB)+ xBln(xB)) + EGm
• EGm = xAxABLA,AB + …
• When the liquid is pure AB the
configurational entropy is almost zero
Quasichemical model
• Quasichemical models are derived using mixing
the fractions of bonds yAA, yAB and yBB rather
than constituents yA and yB. But one may also
treat this as a model with the additional
constituents AB and BA and a quasichemical
configurational entropy.
• Sm = -Rz/2 (yAAln(yAA/yAyA) + yABln(yAB/yAyB)
+ yBAln(yBA/yByA) + yBBln(yBB/yByB))
- R(yAln(yA)+yBln(yB))
01003-p.21
Quasichemical model
• There is a reason to have both AB and BA bonds
as in a lattice this is related to the constituents to
the left or right of the bond. The fraction of the
constituents can be calculated from the ”bond”
fractions
y’A = 0.5(yAA + yAB)
y’B = 0.5(yBA + yBB)
y”A = 0.5(yAA+yBA)
y”B = 0.5(yAB + yBB)
• It is possible to include long range order in the
quasichemical model by allowing y’A and y”A to
be different, i.e. yAB not equal to yBA. This is
similar to a lro model (A,B)(A,B)
Quasichemical model
• The degree of short range order, , can be
evaluated from the difference between the
”fraction of bonds” and the product of the
constituent fractions
yAA = y’Ay”A - yAB = y’Ay”B + yBA = y’By”A + yBB = y’By”B - 01003-p.22
Quasichemical model
• Advantages: Physically correct behaviour
when the liquid becomes ideal. Rather
simple to handle.
• Disadvantages: The value of z is not known
in the liquid and it is complicated to let it
vary with composition. Many different
excess models are used.
Ionic liquid model
The ionic liquid model one starts from the
rather unphysical assumption that one has
long range order in the liquid. This may seem
very drastic but for an ionic liquid one may
consider distributing the anions and cations
separately at their positions in space. It would
be very unlikely (require a lot of energy) to
place a cation in a place where there has been
an anion.
01003-p.23
Ionic liquid model
• The first ionic liquid model was proposed by
Temkin 1945 for molten salts. It assumed one
sublattice for anions and one for cations
(C+c)P(A-a)Q . P and Q must vary with
composition to ensure electroneutrality.
• Instead of using P and Q one can define
”equivalent fractions” zA as follows
• zA= vAxA/(vAxA+ vBxB+…)
where vA is the charge of ion A and xA the
mole fraction of A
Ionic liquid model
• Many liquids change in character with
composition. For example Cu-Fe-S has
metallic behaviour along Cu-Fe but ionic
behaviour along Cu-S and Fe-S. A model
to describe this ternary must handle both
these behaviours.
01003-p.24
Ionic liquid model
• In 1984 Hillert extended the Temkin model
by adding hypothetical vacancies to the anion
sublattice. It was also allowed to have neutral
species there. But the cation sublattice could
only contain cations. (C+c)P(A-a,Va,K0)Q
• P and Q vary with composition and are
calculated as the average charge on the
opposite sublattice.
• The vacancies are assumed to have an
”induced” charge equal to Q.
Ionic liquid model
• The new model described metallic subsystems
as a regular solution by having only Va in the
anion sublattice (Fe+2,Cu+1)P(Va)Q
• Systems like Fe-C where C is assumed to have
no tendency to take or give electrones can be
modelled as (Fe+2)P(Va,C)Q which turns out to
be identical to a regular solution model Fe-C.
01003-p.25
Ionic liquid model
• Most surprisingly the ionic model for Fe-S,
(Fe+2)P(S-2,Va,S0)Q turned out to be identical
to the associated model (Fe, FeS,S)
Ionic liquid model
• For oxides and other systems the ionic liquid
model has been used extensivly, for example
Ca-Fe-Si-O modelled as
(Ca+2,Fe+2,Si+4)P(O-2,SiO4-4,Va,SiO2,FeO1.5)Q
01003-p.26
Ionic liquid model
• Advantages: Can handle all types of liquid
phases (except aqueous solutions)
• Disadvantages: The model has a strong
tendency to form reciprocal miscibility gaps
that are difficult to control.
Model selection
• Special behaviour of data, like strong ”V” shaped
enthalpy of mixing (sharp raise of activity). In a
crystalline phase this indicates long range order, in a
liquid short range order.
• Same phase may occur in several places in the
system (or related phases like ordered superstructures)
• Use same models as in previous assessments of the
same phase.
01003-p.27
Model selection
• The terminal phases are those for the pure
elements, ”end-members”. Usually the properties
of these are known for both components (lattice
stabilities). One must not change these lattice
stabilities without a very good reason.
• The mixing in a terminal phase may be
substitutional or interstitial.
• Sublattice phases may have different mixing on
each sublattice.
Model selection
• Some terminal phases may extend far into the
system and this makes it possible to determine the
excess Gibbs energy accurately. But often the
solubility is very low and one must use few excess
parameters.
• In some cases one may have the same terminal
phase on both sides but one or more intermediate
phases in between. But the terminal phases must
be modelled as the same phase.
01003-p.28
2700
2700
2400
2400
TEMPERATURE_CELSIUS
TEMPERATURE_CELSIUS
The Fe-Mo with and without
intermediate phases
2100
1800
1500
1200
900
600
2100
1800
1500
1200
900
600
300
300
0
0
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 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 MO
MOLE_FRACTION MO
Model selection
• An intermediate phase is any phase which does not
extend to the pure elements (or end-members).
• The name intermediate is preferred to intermetallic as
it can also be used for carbides and oxides.
• An intermediate phase can have extensive solubility
and a simple lattice like BCC or FCC but often it has
restricted solubility and sometimes the structure is
unkown. Most often they have at least two sublattices
with different constituents. The Compound Energy
Formalism (CEF) can often provide a reasonable
model.
01003-p.29
Cu-Zn, several BCC phases
Intermediate phases
• These phases usually have different types of
sites with different elements entering these
sublattices. For example the phase
modelled with 3 sublattices with 10:4:16
sites and with ”fcc” elements on the first,
”bcc” elements on the second and all
elements on the third.
• Example
(Co,Fe,Ni)10(Cr,Mo)4(Co,Cr,Fe,Mo,Ni)16
01003-p.30
Intermediate phases
• An inital set of parameters for an intermediate phase
with sublattices are using the GHSERxx functions like
• G(MU,FE:MO:FE)=7*GFEFCC + 2*GHSERMO +
4*GHSERFE+V1+V2*T
• V1 is a heat of formation and V2 is an entropy of
formation.
• With no Tln(T) term one assumes Kopp-Neuman rule
for the heat capacity. Only if there are experimental
heat capacities one may use a Tln(T) term
• It is often enough to use just the G parameters for
intermediate phases, one should avoid using
interaction parameters.
Intermediate phases
• Laves phases have the stoichiometry A2B and there are
several different called C14, C15 and C36.
Additionally there are other phases with the same
stoichiometry that are not Laves phases.
• It has been a general agreement to model the Laves
phases as (A,B)2(B,A) assuming that the only defects
are anti-sites atoms.
• There has also been a general agreement that the pure
elements in a Laves phase are given as
G(Laves,A:A)=3*GHSERAA+15000
• Today with ab initio data this may change.
01003-p.31
Intermediate phases
• Carbides and nitrides are usually modelled with
one sublattice for the metals and one for the
carbon or nitrogen. But sometimes there are
several sublattices for the metals.
• The fcc carbonitride, like TiC or VN is modelled
as an FCC phase with most of the interstitial sites
filled with carbon or nitrogen. This is a second
composition set.
• The hcp carbonitride, like M2C or M2N is also
modelled as a second composition set of the HCP
phase
Phases with order/disorder transformations
• Some intermediate phases represent ordered BCC,
FCC and HCP lattices. There are a number of
established techniques to model phases with
order/disorder as the ordered phases should become
identical to the disordered phase when the ordering
is no longer stable. This is achieved by partitioning
the Gibbs energy in two parts
• Gm = Gmdis + Gmord
Gmord is zero when the phase is disordered
01003-p.32
Advantages with partitioning
• Simpler assessment of binaries and ternaries
as the ordering part can be assessed
independently of the disordered phase
• Simpler merging with other systems where
the phase is only disordered. Those
parameters are just added to the disordered
part.
Order/disorder
• A2/B2 ordering in BCC is modelled with two
sublattices, sometimes also with a third for
interstitials. (Fe,Al,Ni)0.5(Fe,Al,Ni)0.5(Va,C)3
• B2 ordering require G(B2,Fe:Al)=G(B2,Al:Fe) as
the two sublattices are identical. The interaction
parameters on both sublattices must also be equal.
• A1/L12 ordering is often modelled with two
sublattices (or 3 with interstitials)
(Ni,Al,Fe)0.75(Al,Fe,Ni)0.25(Va,C)1
• This model has many complicated relations
between the parameters.
01003-p.33
A1/L12 and A2/B2
FCC with A1 and L12 ordering (A,B)0.75(A,B)0.25
BCC with A2 and B2 ordering (A,B)0.5(A,B)0.5
Order/disorder
• A better description of ordering in FCC_A1 is
obtained with four sublattices. In this case
both L12 and L10 ordering can be described.
The relations between the parameters are also
simpler as all nearest neighbours are on
another sublattice.
• Ordering in HCP_A3 using CEF is treated
identically with ordering in FCC_A1.
01003-p.34
Order/disorder for BCC
• A2/B2 ordering in BCC is modelled with
two sublattices, sometimes also with a third
for interstitials.
(Fe,Al,Ni)0.5(Fe,Al,Ni)0.5(Va,C)3
• B2 ordering require
G(B2,Fe:Al)=G(B2,Al:Fe) as the two
sublattices are identical. All interaction
parameters on both sublattices must also be
equal, for example
L(B2,Al,Fe:Al:Va)=L(B2,Al:Fe,Al:Va).
The BCC and B2 structures
All nearest neighbour bonds are between
elements in different sublattices
01003-p.35
Order/disorder for FCC and HCP
• A1/L12 ordering is often modelled with two
sublattices (or 3 with interstitials)
(Ni,Al,Fe)0.75(Al,Fe,Ni)0.25(Va,C)1
• This model has many complicated relations
between the parameters because there are
nearest neigbour bonds both between the
sublattices and within the sublattice with
0.75 sites.
Constitutional square A3B1
A3B
1.0
B3B
0.9
0.8
0.7
Y
0.6
0.5
0.4
0.3
0.2
0.1
0
0
A3A
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
X
B3A
01003-p.36
The L12 structure
In a 2 sublattice model the red atoms have 8
nearest neighbours one the same sublattice and 4
nearest neigbours on the other sublattice.
Order/disorder for FCC
• A better description of ordering in FCC_A1 is
obtained with four sublattices. In this case
both L12 and L10 ordering can be described.
The 4 sublattices represent a tetrahedron in
the lattice. The relations between the
parameters are also simpler as all nearest
neighbours are on another sublattice.
• Ordering in HCP_A3 is treated identically
with ordering in FCC_A1 with 4 sublattice
CEF.
01003-p.37
FCC ordering without SRO
200
TEMPERATURE_KELVIN
180
160
140
120
100
80
60
40
20
0
0
0.1
0.2
0.3
0.4
0.5
MOLE_FRACTION B
FCC ordering with CVM and CEF
100
100
90
TEMPERATURE_KELVIN
TEMPERATURE_KELVIN
90
80
70
60
50
40
30
80
70
60
50
40
30
20
20
10
10
0
0
0
0.1
0.2
0.3
0.4
0
0.5
0.1
0.2
0.3
MOLE_FRACTION B
MOLE_FRACTION B
01003-p.38
0.4
0.5
FCC ordering in Au-Cu
FCC ordering with 4 sublattices
• All parameters G(ORDFCC,A:A:A:B) must be
equal independently which sublattice with B.
• All interaction parameters like
L(ORDFCC,A,B:A:A:A) must be equal
independently which sublattice have interaction.
• The 4 sublattice model is a tetrahedron in the FCC
lattice and GA:A:A:B has 3 bonds between A and B.
• GA:A:B:B has 4 bonds between A and B
01003-p.39
More about FCC ordering
• In a simple bond energy model when the
bond between A and B is independent of the
composition we would thus have
• GA:A:A:B = 3uAB
• GA:A:B:B = 4uAB
• GA:B:B:B = 3uAB
• It is possible to calculate metastable G by
first principle techniques.
More about FCC ordering
• In CEF there is no explicit short range order (sro)
but it is possible to show that a first approximation
of the contribution to the Gibbs energy from sro
can be modelled by the reciprocal parameters
LA,B:A,B:A:A
• If these cannot be fitted to experimental data one
can use the approximation LA,B:A,B:A:A = uAB
01003-p.40
Ordering in ternaries
A third component added to an ordered phase
many stabilize or destabilize the ordering. If the
third component does not form this ordered
phase it may be necessary to use first principle
calculations to determine if the component
would enhance the ordering or not.
Al-Ni-Ti at 1273 K
0.50
AC
TI O
NA
L
0.45
0.40
0.35
WE
IG
HT
_F
R
0.30
0.25
0.20
H_L21
0.15
0.10
L12
0.05
FCC
0
0
DO24
0.1
0.2
0.3
0.4
WEIGHT_FRACTION TI
01003-p.41
0.5
Some other metal-nonmetal phases
• Examples of such phases are carbides,
nitrides, oxides, sulphides etc.
• Unless the constituents are charged one can
model these with a normal CEF. In the
simplest case the metals occupy one
sublattice and the non-metal another like in
M7C3. In other cases more sublattices are
needed.
Some other metal-nonmetal phases
• Some carbides, nitrides etc have the same
structure as some metallic phases, like the
cubic carbonitride (Nb,Ti,V,…)(C,N,Va)
which has the same structure as the
interstitial solution of C or N in austenite.
01003-p.42
Some other metal-nonmetal phases
• If the phase has charged constituents, ions, one
have to consider defects and the charge balance.
For wustite, approximately FeO, some Fe is
always trivalent and the model to use is
(Fe+2,Fe+3,Va)1(O-2)1.
• This model has three end-members, two of which
are charged. Only neutral combinations of
charged end-members have any physical
significance. Note the vacanices are neutral!
• One may have charged vacancies in a solid phase
representing electrones or holes.
Halite (wustite) modelled as
(Fe+2, Fe+3, Va)1(O-2)1.
The line represent the neutral combination of
constituents
1.0
0.9
0.8
Y(H
AL
,FE
+3
)
0.7
0.6
0.5
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
Y(HAL,VA)
01003-p.43
Halite parameters
• Note the end-member G(halite,Va:O-2).
This parameter should be set to zero as it
will appear in all halite phases, not just
wustite.
• For ionic phases one arbitrary parameter
can be set to zero as that represent the
energy of creating an ion.
Some other metal-nonmetal phases
1.0
1:T,Y(SPINEL,FE+3)
2:T,Y(SPINEL,FE+3#2)
3:T,Y(SPINEL,VA#2)
1
0.9
0.8
1
0.7
Y(SPIN,*)
More complicated phases like the
magnetite, Fe3O4which is a spinel,
can have several sublattices with
charged constituents
(Fe+2,Fe+3)1(Fe+2,Fe+3,Va)2(O-2)4.
This phas is ”inverse” at low
temperature and ”randon” at high.
2
0.6
2
0.5
0.4
0.3
0.2
0.1
0
500
3
3
1000
1500
T
01003-p.44
2000
Modelling U-O
The phase diagram for
the U-O system has many
modelling challenges.
For the liquid the ionic
liquid has been used. For
the UO2 hase a model
with several charges of
U and vacancy and
oxygen interstitial defects
(U+3,U+4,U+5)1(O-2,Va)2(Va,O-2)1
Modelling U-O
Both the varying valence of U and oxygen vacancies
and interstitials are needed to model the whole
composition range.
At the ideal stoichiometry one may consider
”electronic defects”, i.e. 2U+4 = U+3 + U+5
using the model (U+3,U+4,U+5)1(O-2)2
and that some oxygen may go to the interstitial sites is
modelled according to the reciprocal model
(U+4)1(O-2, Va)2(Va, O-2)1
01003-p.45
Modeling U-O
Using the sublattice model for these kind of defects
gives a much more flexible model than a ”normal”
Wagner-Schottky model.
It can also be easily extended to include more
elements like Pu, Am, Np and other, more or less,
nice elements.
Ab initio values for the defect energies can be used
in the assessment together with all other data.
Conclusions
• The CALPHAD method has a well established set of tools
for modelling almost any kind of thermodynamic system.
• Many popular models like ”dilute solutions” and ”WagnerSchottky” have much more powerful equivalents inside the
CALPHAD method.
• Thermodynamic models are the only method available
today to do truely multicomponent modelling over wide
composition, temperature and pressure ranges
• Other properties like magnetism, mobilities, surface
energies etc. can be modelled consistently in the same way.
01003-p.46
End of lecture
01003-p.47