Lecture OpenCalphad OC-2014

A free software for thermodynamic calculations

Bo Sundman1, Ursula R Kattner2, Mauro Palumbo3 andSuzana G Fries3

1 INSTN CEA Saclay, France2 NIST, Gaithersburq, MD, USA

3 ICAMS, Ruhr University Bochum, Germany

March 7, 2014

Abstract

The use of thermodynamics in many applications suffers from thelack of a high quality open source software for multicomponentsystems. The aim of the OC software is to fill this gap.

TMS 7 March 2013, Bo Sundman et al.

Abstract

The use of thermodynamics in many applications suffers from thelack of a high quality open source software for multicomponentsystems. The aim of the OC software is to fill this gap.It is also very important for the development of thermodynamicdatabases to improve the thermodynamic models implemented inthe software. An open source software with a GNU license willmake it easier for those interested to develop and test new models.


Content

◮ Data structures

◮ Software for storage, retrieval and calculation of the Gibbsenergy for a single phase, the model package.

◮ Software for calculation of equilibria for various conditions.

◮ User interface.

◮ Assessment of experimental and theoretical data forthermodynamic databases.

◮ Software library for application programmes like phase fieldsimulations.


The elements

A thermodynamic system is buit fromelements. Vacancies and electrons areincluded.


The elements and species

A thermodynamic system is buit fromelements. Vacancies and electrons areincluded.The species are stoichiometriccombinations of elements.


The elements, species and phasesA thermodynamic system is buit fromelements. Vacancies and electrons areincluded.The species are stoichiometriccombinations of elements.

The phases have species asconstituents. The phases can havedifferent models, constituents,sublattices, magnetic contribution.


The elements, species and phasesA thermodynamic system is buit fromelements. Vacancies and electrons areincluded.The species are stoichiometriccombinations of elements.

The phases have species asconstituents. The phases can havedifferent models, constituents,sublattices, magnetic contribution.Model parameters describe the Gibbsenergy but also other compositiondependent properties like Curietemperature, elastic constants,mobilities, etc.


Parameter tree for a phase (A,D)(B,C)

The parameters for a phase are stored in a tree structure.



The parameters for a phase are stored in a tree structure.The endmember records specify one constituent in each sublatticeand has a property record list which contain an expression of theGibbs energy and maybe other things like the Curie temperature,mobilities etc. for this endmember.



The parameters for a phase are stored in a tree structure.The endmember records specify one constituent in each sublatticeand has a property record list which contain an expression of theGibbs energy and maybe other things like the Curie temperature,mobilities etc. for this endmember.Each interaction record containsone more constituent in a sublattice and has also a property list.



The parameters for a phase are stored in a tree structure.In fact the data structure gives the model equation:

Gm = yAyB ( ◦GA:B + yC (LA:B,C + yDLA,D:B,C) + yDLA,D:B) + · · ·


Static and dynamic data

The data for elements, speciesand model parameters areindependent on the externalconditions on the system


Static and dynamic data

The data for elements, speciesand model parameters areindependent on the externalconditions on the system

Dynamic data

The amount and constitution ofthe phase varies with the externalconditions like temperatue andcomposition and are stored in theequilibrium record.


Parallell execution

Dynamic data

In addition to the dynamic phasedata the current conditions arealso stored in the equilibriumrecord. In simple cases there isjust one equilibrium record buteach is independent and can beexecuted in parallel.


Parallell execution

Dynamic data

In addition to the dynamic phasedata the current conditions arealso stored in the equilibriumrecord. In simple cases there isjust one equilibrium record buteach is independent and can beexecuted in parallel.This is useful for applicationsoftware like mapping phasediagrams, assessment of modelparameters using experimentaland theoretical data and forphase field simulations wheneach gridpoint can have its ownset of conditions.


Models of phases

◮ The simplest model is an ideal substitutional solution of theelements. The elements have just a Gibbs energy of formationfor the phase: ◦Gα

A, relative to a reference state.


Models of phases



◮ The next step is to have an ideal mixing of species asconstituents, like in an ideal gas. Each species I has a Gibbsenergy of formation: ◦Gα

i .


Models of phases




i .

◮ The next step is to introduce interactions between theconstituents, i.e. a regular solution. The interaction candepend on the temperature, T , and the constitution, Y .


Models of phases




i .

◮ The next step is to introduce interactions between theconstituents, i.e. a regular solution. The interaction candepend on the temperature, T , and the constitution, Y .

◮ For crystalline phases one must take long range ordering intoaccount using sublattices for example in interstitial solutions,Laves, σ etc. For such models we can introduce the Gibbsenergy of formation of an “end-members” with oneconstituent in each sublattice, like ◦Gσ

Fe:Cr:Cr:Fe:Cr for a 5sublattice model of the σ phase.


The Compound Energy Formalism, CEF

The Compound Energy Formalism (CEF) contains all these models(and many more) as special cases. In addition the constituents canbe charged like in a spinel phase and one can model the magneticcontribution to the Gibbs energy as a separate function dependingon the Curie temperature and the Bohr magneton number. Thegeneric formula of CEF for a phase is

Gm = srfGm − T cfgSm + EGm + physGm (1)

The term physGm can describe special contributions to the Gibbsenergy for example ferromagnetic transitions.The CEF model has been implemented in the OC software.


Software, modules

The OC software is modulized in the following way. Each moduledepends on the more basic ones.

◮ Application programs using the OC application softwareinterface. The assessment module is a special applicationintegrated in the OC software.


Software, modules



◮ Module for calculation and plotting of diagrams.


Software, modules




◮ Module for single equilibrium calculation for the flexibleexternal conditions on T ,P , overall composition, chemicalpotentials, specification of fix phases etc. This module mustiterate by varing the amount of the phases and theirconstitutions to find the minimum for the given set ofconditions.


Software, modules




◮ Module for single equilibrium calculation for the flexibleexternal conditions on T ,P , overall composition, chemicalpotentials, specification of fix phases etc. This module mustiterate by varing the amount of the phases and theirconstitutions to find the minimum for the given set ofconditions.

◮ Module with different thermodynamic models to calculate themolar Gibbs energy and partial derivatives with respect toT ,P ,Y for each phase in the system.


Running an OC macro to calculate an equilibrium in the C-Cr-Fesystem:OC:macro ocex02bOC:read tdb ccrfe

reading a TDB file

Created records for elements, species, phases: 3 11 9

end members, interactions, properties: 23 16 57

TP-funs, composition sets, equilibria: 80 10 1

state variable functions, references, additions: 3 7 2

OC:set c t=1000 p=1e5OC:set c n(fe)=.75 n(c)=.1 n(cr)=.15OC:c e

Entering calceq2

Composition set(s) created: 1

Grid minimization: 167 gridpoints 0.0000E+00 s and 16 clockcycles

Done grid minimization, 3 stable phase(s):

Too soon to remove phase 8 3 0

Not added comp.set with same composition as stable phase: 8 1 2

Solution found after 19 iterations.

Equilibrium calculation 1.5600E-02 s and 32 clockcycles


OC:list res 1

Conditions .................................................:

1: T=1000, 2: P=100000, 3: N(FE)=0.75, 4: N(C)=0.1, 5: N(CR)=0.15

Degrees of freedom are 0

Some global data ...........................................:

T= 1000.00 K ( 726.85 C), P= 1.0000E+05 Pa, V= 0.0000E+00 m3

N= 1.0000E+00 moles, B= 5.0886E-02 kg, RT= 8.3145E+03 J/mol

G= -4.3287E+04 J, G/N= -4.3287E+04 J/mol, H= 2.1653E+04 J, S= 6.4940E+01 J/K

Some component data ........................................:

Component name Moles Mole-fr Chem.pot/RT Activities Ref.state

C 1.0000E-01 0.10000 -2.6108E+00 7.3478E-02 SER (default)

CR 1.5000E-01 0.15000 -7.4883E+00 5.5957E-04 SER (default)

FE 7.5000E-01 0.75000 -5.0959E+00 6.1218E-03 SER (default)

Some Phase data ............................................:

Phase: BCC_A2 Status: Entered Driving force: 0.0000E+00

Formula units: 6.6716E-01, Moles of atoms/formula unit: 1.0003E+00

Moles 6.6737E-01, Mass 3.7232E-02 kg, Volume 0.0000E+00 m3, Mole fractions:

FE 9.88074E-01 CR 1.16085E-02 C 3.17582E-04

Phase: M7C3_AUTO#2 Status: Entered Driving force: 0.0000E+00



CR 4.27665E-01 C 3.00000E-01 FE 2.72335E-01


OC:set status phase liquid=fix 0OC:set c t=noneOC:c eOC:list,,,,,

Conditions .................................................:

1: P=100000, 2: N(FE)=0.75, 3: N(C)=0.1, 4: N(CR)=0.15, 5: <LIQUID>=0

Degrees of freedom are 0

Some global data ...........................................:

T= 1536.30 K ( 1263.15 C), P= 1.0000E+05 Pa, V= 0.0000E+00 m3

N= 1.0000E+00 moles, B= 5.0886E-02 kg, RT= 1.2774E+04 J/mol

G= -8.4166E+04 J, G/N= -8.4166E+04 J/mol, H= 4.5920E+04 J, S= 8.4675E+01 J/K

Some component data ........................................:

Component name Moles Mole-fr Chem.pot/RT Activities Ref.state

C 1.0000E-01 0.10000 -3.9252E+00 1.9739E-02 SER (default)

CR 1.5000E-01 0.15000 -7.7362E+00 4.3671E-04 SER (default)

FE 7.5000E-01 0.75000 -6.7148E+00 1.2128E-03 SER (default)

Some Phase data ............................................:

Phase: LIQUID Status: Fixed Driving force: 0.0000E+00

Formula units: 0.0000E+00, Moles of atoms/formula unit: 1.0000E+00

Moles 0.0000E+00, Mass 0.0000E+00 kg, Volume 0.0000E+00 m3, Mole fractions:

FE 6.84186E-01 CR 1.69723E-01 C 1.46091E-01

Phase: FCC_A1 Status: Entered Driving force: 0.0000E+00



FE 8.36622E-01 CR 1.02347E-01 C 6.10314E-02

Phase: M7C3 Status: Entered Driving force: 0.0000E+00



CR 3.94573E-01 FE 3.05427E-01 C 3.00000E-01


Thermodynamic databases

For any application of computational thermodynamics (CT) onemust have a database with the model parameters. In the simplestcase one can just use results from DFT calculations of theenthalpies of formation of the different phases.



For any application of computational thermodynamics (CT) onemust have a database with the model parameters. In the simplestcase one can just use results from DFT calculations of theenthalpies of formation of the different phases.Calculations of equilibria require that the energy parameters havean accuracy of a few joule (less than 0.1 meV) and thuscalculations based only on DFT results can be very far from theexperimental information. But the assessment procedure in CTmakes it possible to combine DFT reults with experimental data toobtain model parameters with a good fit to reality.



For any application of computational thermodynamics (CT) onemust have a database with the model parameters. In the simplestcase one can just use results from DFT calculations of theenthalpies of formation of the different phases.Calculations of equilibria require that the energy parameters havean accuracy of a few joule (less than 0.1 meV) and thuscalculations based only on DFT results can be very far from theexperimental information. But the assessment procedure in CTmakes it possible to combine DFT reults with experimental data toobtain model parameters with a good fit to reality.Most high quality databases are commercial but there are a fewsmaller databases available free. However, you must be carefulusing databases that may not be maintained and updated by anexperienced and skillfull assessment team.


Single equilibrium calculation

To find the equilibrium at fixed temperature, pressure andcomposition one must minimize the total Gibbs energy of thesystem. The models describe the molar Gibbs energy, Gα

m and thetotal energy is:

G =∑

α

ℵαGα

m (2)

where ℵα is the amount of the phases.


Single equilibrium calculation

To find the equilibrium at fixed temperature, pressure andcomposition one must minimize the total Gibbs energy of thesystem. The models describe the molar Gibbs energy, Gα

m and thetotal energy is:

G =∑

α

ℵαGα

m (2)

where ℵα is the amount of the phases.

If one has other conditions like fixed volume, chemical potentials orfixed phases one can modify the Gibbs energy by the use ofLagrangian multipliers.


Single equilibrium calculation, Lagrangian method

The algorithm for minimization that is implemented in OC wasproposed by Hillert (1981). It uses Lagrangian multipliers to takecare of both external and interal constraints. For constantcomposition and fixed T and P an example is:

L = G +∑

A

(NA −

∑

α

ℵαMα

A)µA +∑

α

∑

s

ηα,(s)(∑

i

yα,(s)i − 1) (3)

where NA is the total amount of component A, MαA is the amount

of A in α, yi is the constituent fraction of i . µA and η areLagrangian multipliers.


Single equilibrium calculation, Lagrangian method

The algorithm for minimization that is implemented in OC wasproposed by Hillert (1981). It uses Lagrangian multipliers to takecare of both external and interal constraints. For constantcomposition and fixed T and P an example is:

L = G +∑

A

(NA −

∑

α

ℵαMα

A)µA +∑

α

∑

s

ηα,(s)(∑

i

yα,(s)i − 1) (3)

where NA is the total amount of component A, MαA is the amount

of A in α, yi is the constituent fraction of i . µA and η areLagrangian multipliers.The first constraint is the massbalance and the second that thesum of constituent fractions in each sublattice of α is unity.By differentiating the Lagrangian function with respect to thedifferent variables we obtain many relations.


Single equilibrium calculation module, step 1

In a first step we invert the matrix of second derivatives of theGibbs energy with respect to the constituent fractions:

eαij =

(

∂2Gαm

∂yi∂yj

)

−1

(4)




eαij =

(

∂2Gαm

∂yi∂yj

)

−1

(4)

This matrix is used to formulate an equation for correcting theconstituent fractions, ∆yαi :

∆yαi =∑

A

∑

j

∂MA

∂yjeijµA −

∑

j

∂Gm

∂yjeij −

∑

j

∂2Gm

∂T∂yjeij∆T (5)

where µA the chemical potential of component A. If T is constantthe ∆T term cancels. All the partial derivatives are calculatedanalytically in the model module.




eαij =

(

∂2Gαm

∂yi∂yj

)

−1

(4)

This matrix is used to formulate an equation for correcting theconstituent fractions, ∆yαi :

∆yαi =∑

A

∑

j

∂MA

∂yjeijµA −

∑

j

∂Gm

∂yjeij −

∑

j

∂2Gm

∂T∂yjeij∆T (5)

where µA the chemical potential of component A. If T is constantthe ∆T term cancels. All the partial derivatives are calculatedanalytically in the model module. But to solve this we must firstfind µA and ∆T which is done in a second step.



In the second step we formulate a system of linear equations toobtain a new values of the chemical potentials, amounts of thestable phases and temperature if it is variable. There is oneequation for each stable phase like

∑

A

MαAµA = Gα

m (6)

if any chemical potential is fixed that term is moved to the righthand side.



In the second step we formulate a system of linear equations toobtain a new values of the chemical potentials, amounts of thestable phases and temperature if it is variable. There is oneequation for each stable phase like

∑

A

MαAµA = Gα

m (6)

if any chemical potential is fixed that term is moved to the righthand side. For each condition on an extensive variable we obtainan equation, which in a simple case, NA = fix , is:

∑

α ℵα∑

i

∂Mα

A

∂yi

∑

B

∑

j

∂Mα

B

∂yjeαij µB +

∑

α MαA∆ℵ

α =∑

α ℵα∑

i

∑

j

∂Mα

A

∂yi

∂Gα

m

∂yjeαij

where we find the matrix eij again.



With the new values of the chemical potentials and amounts ofphases, we can obtain new constitutions of the phases using eq. 5.



With the new values of the chemical potentials and amounts ofphases, we can obtain new constitutions of the phases using eq. 5.If a phase amount becomes negative that phase should be removedfrom the set of stable phases and if the driving force, ∆G

∆Gβ =∑

A

MβAµA − Gβ

m (7)

for an unstable phase becomes positive that phase should be addedto the stable set of phases.



With the new values of the chemical potentials and amounts ofphases, we can obtain new constitutions of the phases using eq. 5.If a phase amount becomes negative that phase should be removedfrom the set of stable phases and if the driving force, ∆G

∆Gβ =∑

A

MβAµA − Gβ

m (7)

for an unstable phase becomes positive that phase should be addedto the stable set of phases.If the change in chemical potentials and phase amounts are smalland all external conditions are fullfilled the calculation hasconverged, otherwise we start again from step 1.


Grid minimizer, step 0

The algorithm described is an iterative method which dependsstrongly on the initial configuration of the phases.


Grid minimizer, step 0

The algorithm described is an iterative method which dependsstrongly on the initial configuration of the phases.As an initial step 0 the OC software has a grid minimizer whichcalculates the Gibbs energy of all phases over a grid ofcompositions. These are treated as stoichiometric phases in apreliminary minimization to find a set of gridpoints representingthe minimum. These gridpoints are used as initial guess and theircompositions are inserted as start constitutions for the interativealgorithm.


Grid minimizer 2

Two phases in a binary system can have Gibbs energy curves at afixed temperature and pressure like in (a). If the initalcompositions of these phase are as marked in (b) an iterativeminimizer, for a composition in the middle in the system, will findthe equilibrium marked with the common tangent in (c).

-1

0

1

2

3

4

5

6

7

Gib

bs e

nerg

y (k

J/m

ol)

0 0.2 0.4 0.6 0.8 1.0

X(B)

-1

0

1

2

3

4

5

6

7 G

ibbs

ene

rgy

(kJ/

mol

)

0 0.2 0.4 0.6 0.8 1.0

X(B)

-1

0

1

2

3

4

5

6

7

Gib

bs e

nerg

y (k

J/m

ol)

0 0.2 0.4 0.6 0.8 1.0

X(B)

-1

0

1

2

3

4

5

6

7

Gib

bs e

nerg

y (k

J/m

ol)

0 0.2 0.4 0.6 0.8 1.0

X(B)

. (a) (b) (c) (d)The real equilibrium is the common tangent marked in (d)


Grid minimizer 3

With a grid minimizer one calculates the Gibbs energy along thecurves at fixed compositions like in (e). These are treated asindividual phase with fix composition as in (f) in a firstminimization and one will find the two gridpoints representing thelowest Gibbs energy in (g).

-1

0

1

2

3

4

5

6

7

Gib

bs e

nerg

y (k

J/m

ol)

0 0.2 0.4 0.6 0.8 1.0

X(B)

-1

0

1

2

3

4

5

6

7 G

ibbs

ene

rgy

(kJ/

mol

)

0 0.2 0.4 0.6 0.8 1.0

X(B)

-1

0

1

2

3

4

5

6

7

Gib

bs e

nerg

y (k

J/m

ol)

0 0.2 0.4 0.6 0.8 1.0

X(B)

-1

0

1

2

3

4

5

6

7

Gib

bs e

nerg

y (k

J/m

ol)

0 0.2 0.4 0.6 0.8 1.0

X(B)

. (e) (f) (g) (h)Using these compositions as start points in an interativeminimization one will find the global minimim ib (h).


On the list to do

The equilibrium calculation is currently not very stable, itfrequently fails and sometimes even converges to the wrongequilibrium. This is mainly due to problems changing the set ofstable phases and to solve these problems we need a larger groupof testers who have patience enough to use the software and senderror reports.


On the list to do


◮ For the future the next thing is to develop the step and mapprocedures for various types of diagrams.

◮ and after that to develop an assessment module to fit modelparameters to experimental and theoretical data.


On the list to do


◮ For the future the next thing is to develop the step and mapprocedures for various types of diagrams.

◮ and after that to develop an assessment module to fit modelparameters to experimental and theoretical data.

In parallell more models will also be added, like the ionic liquidmodel and the corrected quasichemical model for liquids.


Other features

◮ The user interface is a rudimentary command interface with aVAX/VMS flavour, a famous computer/operating system 30years ago.


Other features


◮ The code is written in the new Fortran standard.


Other features



◮ The data structures and the code are quite extensivelydocumented.


Other features




◮ A number of sample macro files is supplied.


Other features





◮ An initial rudimentary software library is provided for the usein application programmes.


Other features






◮ Free for non-commercial use.


Other features






◮ Free for non-commercial use.

All of this is available at:

http://www.opencalphad.org


Welcome to develop and use:

Thanks for listening


Documents

Lecture OpenCalphad OC-2014