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Computer Calculations of Plan for concentrated course
Equilibria and Phase Diagrams • 4 lectures, 2 each week.
• Homework, if well done course ready.
Bo Sundman • Written or oral completition if some
homework not up to requirement.
Understanding thermodynamic models • 3 graduate points.
and using them to determine model • Software and manuals can be downloaded
parameters to fit theoretical and from http://www.thermocalc.se
experimental data
Schedule for seminar room Lectures
• Monday 16/10 at 13-15 • 1: Thermodynamics: Equation of State, Gibbs
energy. Regular solution, sublattice model, lattice
• Thursday 19/10 at 13-15 stabilities, enthalpy models, entropy models,
• Monday 23/10 at 13-15 associate models, quasichemical models, CVM,
configurational terms, Monte Carlo. Numerical
• Thursday 26/10 at 13-15 methods for equilibrium calculation
• Articles: Bo J TRITA, Mats CEF, dilute solutions
• Homework: using TC binary module. Running TC
example 2 and some TC multicomponent
examples. Selecting an individual system for
assessment, oxide demo program
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• 2: Experimental and theoretical information,
crystallography, enthapies, activities, phase • 3: Creating the experimental data file. Modelling
diagram, using ab initio. Criteria for selection of intermetallics like sigma, mu and phases with
models for phases in a system. Numerical methods order/disorder transformations like A2/B2,
for assessment. Assessment of binary systems. A1/L12/L10. Partitioning the Gibbs energy.
Use of PARROT. The alternate mode. Creating databases from assessments,
Kaufman/Ansara assessment method calculating extrapolations to multicomponent systems.
metastable regions. Decision on an individual
system to assess. • Articles: Re-W paper, Ringberg paper.
• Articles: Chapter 7.2 from book and some • Homework: TC example 31, Assessment of Co-V
assessment papers
• Homework: TC example 36, testing different
strategies.
• 4: Strategies to find reasonable set of parameters.
Thermodynamics
Modelling liquids: ionic liquid, molten salts,
aqueous etc. Models for solid metal-nonmetal
systems: carbides, nitrides, oxides etc.
• Articles: 84Hil, ionic liquid, some associated • Equation of state, for example ideal gas
model paper, some assessment papers pV = nRT
• Homework: Assessment of an individual system • Equation of state is not suitable for
selected earlier
modelling of composition dependence.
• The ideal gas equation can be integrated to
a Gibbs energy: G = nRT ln(p/p0)
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Thermodynamics Thermodynamics
• Fundamental equations (unary systems) A multicomponent system consists of many
• U(S,V) Internal energy different species (molecules). There are
• H(S,p) = U + pV Entalpy reactions among these, like 2CO+O2=2CO2
• A(T,V) = U – TS Helmholtz energy The components are an irreducible set of the
species.
• G(T,p) = U + pV – TS Gibbs energy
The fraction of components are independent
but not the fraction of species.
Thermodynamics Change of components
• Fundamental equations (multi-component). • When changing to a new set of components
Ni is moles of component i, yj is constituent the chemical potentials of the new set i is
fraction of species j realated to the original set j by
• G(T,p,Ni) = Σi Ni µi(T,p,yj) • µi = Σj bj µj
• dG=Σi µi dNi • where bj are the stoichiometc factors
• 0 = Σi Ni dµi Gibbs-Duhem
• G(T,p,Ni) = Σϕ Nϕ Gϕm(T,p,yj)
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Thermodynamics Thermodynamics
• A homogeneous reaction between species means
• A phase is a part of space that has homogeneous that the reaction takes place inside a phase, for
composition and structure. example 2H2<g>+O2<g>=2H2O<g>
• The term phase can be extended to non-equilibrium • A heterogenuous reaction means that two or more
cases when the structure and the composition may
vary continuously form one place to another. phases are involved, for example
O2<g>+C<s> = CO2<g>
• The term phase will also be used just for a structure,
with arbitrary components. • Some of the problem in understanding
• The term phase will also be used for the gas, liquid thermodynamics is due to the fact that often no
and amorpheous phases which have no structure. distinction is made between these reactions.
Thermodynamic models
• A model for a phase may contain real and
Thermodynamic models
fictitious species. These species, called the
constituents, contribute to the entropy of mixing. • From Stirlings formula per mole of phase
• The ideal entropy of mixing of a phase having the Sm= -R Σi xi ln(xi)
components as constituents comes from Bolzmann • Mole fraction of components, xi = Ni/ΣΝi
S=R ln( Π(Ni)! / (ΣNi)! )
• Constituent fraction, yi, is equal to the
• This can be derived in two ways, either
amount of the constituent divided by the
distributing different atoms on a given set of
lattice points or from the statistical mechanics of total amount of constituents on a lattice. For
an ideal gas. a gas phase each molecule has a constituent
fraction.
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Thermodynamic models Thermodynamic models
• Elements – those from the periodic chart • Each phase is modelled separately
• Species – an element or a combination of • Phases with no compositional variation has
elements that forms an entity, like H2O, CO2, just an expression G(T,p). That is very
Fe+2 simple to handle at low pressures.
• Constituents are the species that exist in a • Phases with a small compositional variation
phase. A constituent can be real or fictitious. can be very difficult to model as one should
• Components is an irreducible subset of the take into account the different types of
species defects that cause the non-stoichiometry
Thermodynamic models
Thermodynamic model
• The temperature dependence of a Gibbs
energy parameter is normally a polynomial • Properties at low temperature, normally below 300
in T, including a TlnT term from the heat K, is normally not modelled.
capacity • The Gibbs energy at low temperature has a
• G = a + bT + cTlnT + dT2 + … complicated T-dependence (Debye model) that is
• Note that the enthalpy, entropy, heat capacity not easy to combine with the higher temperature
etc can be calculated from this G. properties.
• The pressure dependence, except for a • Enthalpy data at 0 K, from ab initio calculations,
are useful for fitting high temperature data
pressure independent volume, is more
complicated and will be discussed later
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Thermodynamic models Crystallographic data
http://cst-www.nrl.navy.mil/lattice/
• Phases with extensive compositional A1 B1 L10 L12
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 A2 B2 D03 L21
families of simpler lattices, like B1 is A1
with interstitials, B2 is ordered A2 etc.
That should be taken into account in the
modelling
Thermodynamic models
Thermodynamic models
• From the thermodynamic models one can
calculate various thermodynamic properties of a • The Gibbs energy per mole for a solution phase is
system, like heat of transformation, chemical normally divided into four parts
potentials, heat capacities etc • Gm = srfGm – T cfgSm + EGm + physGm
• One may also calculate the phase diagram or srfG
• is the surface of reference for Gibbs energy
metastable extrapolations of the phase diagram m
• cfgS is the configurational entropy
m
• One may make more reliable extrapolations EG is the excess Gibbs energy
in temperature and composition than if one • m
• physG is a physical contribution (magnetic)
extrapolated a single property m
• They can be used in software for simulations
of phase transformations
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Thermodynamic models Regular solutions
• Regular solution models are based on ideal
Modelling by physicists have mainly entropy of mixing of the constituents. In the
concentrated on finding a good general case these are different from the
configurational entropy (Quasichemical, components and their fraction is denoted yi
CVM, Monte Carlo) to describe main
features or a specific detail. • Gϕm = Σi yi oGϕi + RT Σi yi ln(yi) + EGϕm
• oGϕi is the Gibbs energy of pure constituent i
Modelling among material scientists has
mainly concentrated on finding a good in phase ϕ
excess Gibbs energy to reproduce the • EGm is the excess Gibbs energy
experimental data.
Excess energies Regular solution model
• The excess Gibbs energy for a binary system
• If A and B atoms occupy neighbouring EG = Σ Σ
lattice sites the energy of the AB bond is • m i j>i yi yj Lij
related to that of an AA and a BB bond by • Lij = Σν (yi – yj)ν νLij (Redlich-Kister)
• EAB = εAB - 0.5 (εAA + εBB) • Other types of polynomial are possible but all
• If this energy is negative the atoms like to are identical in the binary case. However,
surround themselves with the other kind of they will differ in ternary extrapolations and
atom. thus the most symmetrical is preferred
• If the energy is positive there is a demixing • Lij = 0.5 z Eij where z is the number of nearest
neighbours.
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1000
900
Redlich-Kister coefficients
2003-03-09 20:20:47.58 output by user bosse from GIBBS
TEMPERATURE_KELVIN
800
700
600
500
400
The contribution to
300 the excess enthalpy as
200
0 0.2 0.4 0.6
MOLE_FRACTION Y
0.8 1.0
a function of
composition for the
first three coefficients
Ideal liquid interaction, of the RK series, all
solid interaction 0, with the same value,
+10000 and –10000 10000 J/mol.
Lattice stabilities
Ternary regular solution parameter
A solution model for a phase often extend from
• EGm = yi yj yk L ijk one pure component to another even if one, or
both, of them may not exist as stable in that
• L ijk = vi 0Lijk + vj 1Lijk + vk 2Lijk
phase.
• vi = yi + (1 – yi – yj – yk)/3 These ”lattice stabilities” of the metastable states
of elements was first introduced by Larry
In the ternary system vi = yi. In higher order Kaufman and must be agreed internationally to
systems Σvi = 1 always which guarantes the make assessments compatible. Most commonly
used are those by SGTE, published in Calphad
symmetry.
by Dinsdale 1991
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Liquidus extrapolations for Cr Lattice stability for Cr
First principle calculations have shown fcc-Cr
is mechanically unstable, thus it is impossible
FCC
to calculate the energy difference between fcc
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.
Dilute solution model Sublattice model
• Based on Henry’s law for the activity of the solute Crystalline phases with different types of
and Raoult’s law for the activity of the solvent.
”Epsilon” parameters describe the activity in more
sublattices for the constituents can be
concentrated solutions. described with the sublattice model.
• The assumption that Raoults law is true for Different constituents may enter in the
multicomponent systems is wrong different sublattices and one assumes ideal
• Dilute models are thermodynamically inconsistent entropy of mixing on each sublattice. The
(they do not obey the Gibbs-Duhem equation) and simplest case is the reciprocal system
cannot be used in software for Gibbs energy
minimizations. (A,B)a(C,D)c
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Sublattice model Sublattice model
• The Gibbs energy expression for (A,B)a(C,D)c
Excess Gibbs energy for (A,B)a(C,D)c
• srfGm = Σi Σj y’i y”j oGij
• cfgSm = -R(aΣ i y’i ln(y’i) + cΣ j y”j ln(y”j)) EG
m = y’Ay’B(y”C L A,B:C + y”D L A,B:D) +
• oGij is the Gibbs energy of formation of the
compound iajc , also called “end members”. y”Cy”D(y’AL A:C,D + y”B L B:C,D) +
• a and c are the site ratios y’Ay’By”Cy”D L A,B:C,D
• The excess and physical contributions are as Each L can be a Redlich-Kister series
for a regular solution on each sublattice.
Sublattice model
Sublattice model
• The sublattice model has been used extensively
to describe interstitial solutions, carbides,
• Sublattices are used to describe long range
oxides, intermetallic phases etc.
order (lro) when the atoms are regularly
• It is often called the compound energy
arranged on sublattices over large distances.
formalism (CEF) as one of its features is the
assumption that the compound energies are • Short range order (sro) means that the
independent of composition and it includes fraction of atoms in the neighbourhood of
several models as special cases. an atom deviate from the overall
• Note that the Gm for sublattice phases is usually composition. There are special models for
expressed in moles for formula units, not moles that.
of atoms as vacancies may be constituents.
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Associated solutions
Thermochemical properties for Fe-S
• These are identical to regular solutions except that
one has added one or more fictitious constituents, The phase diagram
for example FeS in liquid Fe-S. The reason to and some thermo-
introduce this is to describe short range order around dynamic properties:
the FeS composition. activity, enthalpy and
• A parameter oGFeS describe the stability of the entropy at 2000 K
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.
Quasichemical model Quasichemical model
• There is a reason to have both AB and BA bonds
• Quasichemical models are derived using mixing
as in a lattice this is related to the constituents to
the fractions of bonds yAA, yAB and yBB rather the left or right of the bond. The fraction of the
than constituents yA and yB. But one may also constituents can be calculated from the ”bond”
treat this as a model with the additional fractions
constituents AB and BA and a quasichemical y’A = 0.5(yAA + yAB) y’B = 0.5(yBA + yBB)
configurational entropy. y”A = 0.5(yAA+yBA) y”B = 0.5(yAB + yBB)
• Sm = -Rz/2 (yAAln(yAA/yAyA) + yABln(yAB/yAyB) • It is possible to include long range order in the
+ yBAln(yBA/yByA) + yBBln(yBB/yByB)) quasichemical model by allowing y’A and y”A to
be different, i.e. yAB not equal to yBA. This is
- R(yAln(yA)+yBln(yB))
similar to a lro model (A,B)(A,B)
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Quasichemical model Cluster Variation Method
• The degree of short range order, ε, can be • An improved method to treat short range order in
evaluated from the difference between the crystalline solids was developed 1951 by Kikuchi
”fraction of bonds” and the product of the and called Cluster Variation Method (CVM). It can
treat arbitarily large clusters of lattice sites but the
constituent fractions entropy expression must be derived for each lattice.
yAA = y’Ay”A - ε • Even for binary systems it can be rather
yAB = y’Ay”B + ε cumbersome to use CVM and for multi-component
systems it is impossible to apply. Anyway, for most
yBA = y’By”A + ε multicomponent phases the contribution to Gibbs
yBB = y’By”B - ε energy due to sro is small.
Comparisons CVM-Associated-Sublattice Numerical consideration
• A model describing short range order should have ideal The Gibbs energy models should be expressed using
entropy of mixing when the pair energy oGAB is zero. the independent constituent variables of the phase.
That is the case for the quasichemical model but not for In some software the mole fraction is used is
the associated model. A reason that the associated independent compostion variable globally and the
model is still used is that it is simpler to handle. constituent fractions (bonds, associates, molecules)
• The relation between the quasichemical model and the of each phase is minimized separately. This is a
sublattice model can be extended to the CVM. A disadvantage as the minimization of the Gibbs
disadvantage with the sublattice model is that it does not energy to find the equilibrium can be made faster for
include sro but it is easier to handle then CVM. the independent constituents of the phase and that no
separate minimization of bond fractions or clusters is
needed.
12
Connection with first principle calculations
Numerical consideration
From first principles one may calculate the energy at 0
Thermo-Calc uses Gibbs energy minimization with K for different configurations of atoms on specific
Lagrangian multipliers. This requires first and second lattices. These energies can be expanded in different
derivatives of the Gibbs energy to have fast and stable ways to describe disordered states for compositions in
convergence. These derivatives are calculated between the calculated configurations. A popular
analytically which require more code but gives faster model to use is the Cluster Expansion Method (CEM)
execution. by Connally-Williams. The cluster energies can then
Although the calculation of second derivatives is not be used in a CVM or Monte Carlo (MC) calculation of
absolutely necessary it has the extra benefit that these the phase diagram for example.
second derivatives are used in the thermodynamic For phase diagram calculations of ordering in binary
factor for the diffusion coefficients and can thus be systems with fcc lattices it is very important to include
used also to speed up simulations. the short range order but for carbides and intermetallic
phases like σ it is less important.
Connections with first principle calculations
The energies from a first principle calculation can also be used Articles
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 • TRITA-MAC about POLY and PARROT
like in the Al-Ni system below some may be metastable and • Description of CEF
their energies must be calculated using ab initio techniques.
• Relation between the dilute solution model
and the regular solution model
13
Homework
End of lecture
• Calculate a number of binary systems using
the BIN module. Select one system and plot
the phase diagram in various ways. Calculate
also G curves and other properties and discuss • See you on Thursday, same time in B2
the relations with the phase diagram.
• Follow TCEX02 and discuss the various
calculations and diagrams.
• Follow TCEX19 and comment on the
calculations.
• Select a binary system you want to assess. It
may be an already assessed system.
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