The CALPHAD method – basis and applications

John ÅgrenMaterials Science and Engineering

KTH (Royal Institute of Technology)SE 100 44 Stockholm, Sweden

The CALPHAD method – basis and applications

The CALPHAD method Webinar 2016-11-15

Content

1. Introduction – A shift in Paradigm!2. General idea behind CALPHAD3. Thermodynamic basis4. Thermodynamic models5. Some Examples: 6. Extension of CALPHAD beyond thermochemistry7. Conclusions


”Materials Genome”Databases and models ICME - Integrated

Computational Materials

EngineeringIntegration of models

Materials designMethod for a purpose

New possibilities: Development time for new materials can be decreased from 10-20 years to 3-4 years.

1. Introduction - A shift in Paradigm!


CALPHAD - the first materials genome because it is ...

• the most efficient way of integrating various pieces of information - different character (thermochemistry, phase diagrams

etc)- coherent and useful form

• extendable far beyond the traditional thermochemistry • increasingly being used outside the traditional

CALPHAD community

A major enabling technology in Materials science and engineering.


CALPHAD – a brief history

• Was born 1970 with the book (Kaufman and Bernstein: Computer calculation of phase diagrams, 1970)

• In the 50:ies and 60:ies Kaufman and Cohen and Hillert had outlined the general principles.

• Early predecessors – van Laar 1908 and later several others.• CALPHAD was introduced as an alternative to the quantum-

based approach PHACOMP (1964).• The annual meetings (Later called CALPHAD conferences) and

the CALPHAD journal both initiated by Kaufman were instrumental for the rapid expansion of the field.




2. General idea behind CALPHADProperties such as • Phase equilibrium

- Phase diagrams- Composition of phases and compounds- Partition coefficients and equilibrium constants...

• Thermochemical data- Heat of reactions and transformations- Heat capacities- Vapour pressures...

• Elastic properties- Bulk modulus- Elastic constants...

• Volume and its thermal expansion...

all stem from a single thermodynamic function of the system of interest.


If that function is known...

then• all these properties may be calculated using that

function.• also other quantities of interest may be calculated

such as- Driving forces for reactions and transformations if the

system is not at equilibrium (e.g. In phase-field simulations)

- Properties of metastable states, e.g. metastable phase diagrams.





• What is the equilibrium state of a system aged under given external conditions?

• What are the driving forces for internal changes if the system is not in equilibrium?

3. Thermodynamic basis


Thermodynamics does not only apply at equilibrium, i.e. the condition when nothing more can change.

• measurements can be performed sometimes quite far outside equilibrium and be extrapolated further into the non-equilibrium regime.

• Thus thermodynamics applies considerably outside equilibrium. But how far outside would it apply? Here we will demonstrate how thermodynamics may be applied as far as needed to solve certain problems.

Important remark:


Overview of thermodynamics

State variables

First law: Energy, heat and work

Second law: Entropy, equilibrium, driving force

Multicomponent systems –chemical potential

External

Internal


System

Internal constitution

SurroundingsExternal conditions: P, V, T, nk …

Interactions between systemand surroundings:

composition Cthermal 1mechanical 1

C + 2

After fixing the external conditionsthe system will approach a state of equilibrium at which there are no further changes.


C + 2 external variables must be fixed inorder to define a unique equilibrium state.

The equilibrium state may be represented by a pointin a C + 2 dimensional space; a state diagram.

A state diagram containing information about what phase is stable is a phase diagram.


Example:1-component system, n = fix, let P and T vary => 2-dimensional state diagram. Add information on phases => unary phase diagram

P

TL

S


Example: Phase diagram of pure iron. Pressure in Pa and temperature in K.


External and internal state variables

External state variablesmay be controlled outside the system by the experimentalist or the processing conditions

Internal state variablesIn a system out of equilibrium additional variables are needed to fully characterize the system. These variables • describe the internal constitution of the system• vary until they reach their equilibrium values and

the system comes to rest


Intensive variablesIf a new system is formed by merging two identicalsystems the values of all intensive variables remainconstant.Ex: P, T, ρ, ck, Hm etc

Extensive variablesIf a new systems is formed by merging two identicalsystems the values of all extensive variables will become twice as large.Ex: V, U, nk, Additative rule:

21 VVV +=


Two types of intensive variables:

- Potentialsex: P, T

- Densities or specific variables =

...k kk k

j

n nρ c xn

= = =∑

extensive variablesize

massvolume volume


The state of equilibrium at given pressure, temperature and composition

Equilibrium for the value of the internal variable ξ that gives the lowest Gibbs energy.


Change in Gibbs energy during a phase transformation.

2:nd law of thermodynamics


a) Metastable equilibrium b) Unstable equilibrium (critical state).

a) b)


Combined 1:st and 2:nd law of thermodynamics in an open system (i.e. Exchange of matter with surroundings), i.e. multicomponent

1

0( , , , )

, , ,

k k ik

i j j

j

j

k j

k j

dU TdS PdV dN Td S

d S D dT

D jd jU U S V NS V N U

= − + −

=

=

=

=

∑

∑driving force for process

Process is frozen in

are natural variables for !

µ

ξ

ξ

ξ

ξ

)law nd:(2

)law (combined

∑

∑≥=

−+−=

jjji

kikk

dDSTd

STddNPdVTdSdU

0ξ

µ


kk

jj

U TSU PVUNU D

µ

ξ

∂=

∂∂

= −∂∂

=∂∂

= −∂

From the combined law:Each derivative tells how much the internal energy changes per unit amount of an extensive quantity, i.e. entropy, volume or number of moles of k, added to the system:Potentials!

Thermal potential (temperature)

Volume potential (negative pressure)

Chemical potential

Reaction potential (negative driving force)


Equilibrium conditions come from second law:

m.equilibriu at reached is which maximum a towards increase

must that implies condition The

:volume constantunder energy constant a with system closed aFor

SSddS

STddNPdVTdSdU

dU

i

kikk

0

0

>=

−+−=

=

∑µ

ξ

S

ξ

S


A more convenient equilibrium condition

m.equilibriu at minimized is energy Gibbs the and given At

m.equilibriu at

energy Gibbs and constant At

:law combined thefrom obtain we and fixed conditions mEquilibriu

GPT

DG

PVTSUGdDPVTSUdPT

dDVdPSdTPVTSUddDPdVTdSdU

PT

TP

0

)(:)(

=−=

∂∂

+−=−=+−

−+−=+−−−=

ξ

ξξ

ξ

Most useful!



Characteristic state functions with their natural variables

),,(),,(),,(),,(),,(

ξξξξξ

VUSPSHPTGVTFVSU

Either one of these functions fully characterize a system.


∑= ∂

∂−

∂∂

+=n

i i

mi

j

mmj

jnm

xGx

xGG

xxxPTG

1

21 )...,,,(

µ

µ

that show to forward straight is It

? is what known, is If


Bx

mG

Example: Binary system – The tangent construction

(1 ) mB m B

B

Gμ G xx

∂= + −

∂

mA m B

B

Gμ G xx

∂= −

∂


1 2 1 2

2

2 2

1 2 1 2

( , , , ... , ...)//

( / ) / /

/...

( , , , ... , ...)

m m

m m

m m

m m

P m

m mk m j

k j

mj

j

m

G G T P x xV G PS G T

G P T G P

c T G T

G GG xx x

GD

G N G T P x x

== ∂ ∂= ∂ ∂

= ∂ ∂ ∂ ∂ ∂

= − ∂ ∂

∂ ∂= + −

∂ ∂

∂= −

∂

=

∑

∑ α α α α α α

α

ξ ξ

α

µ

ξ

ξ ξ

Some properties derived from Gibbs energy of a phase:


Driving force under fixed P, T and composition

.

j jdG D d= −∑Combined law yields:

Consider now a process where the variables change from an initial state to a final state The driving forces may vary in a complicated way during the integration the i

ξ

ξ

finalfinal start

j jstart

dG G G D d= − = −∑∫ ∫

ntegral

The integrated driving force.

ξ


1. Introduction – A shift in Paradigm!2. General idea behind CALPHAD3. Thermodynamic basis4. Thermodynamic models5. Some Examples: 6. Extension of CALPHAD beyond thermochemistry7. Conclusive remarks


If we know Gibbs energy as a function

for the individual phases in a system:

• We may calculate a lot of properties of practical interest.

• Calculate equilibrium state and phase diagram by minimizing G.

• Calculate driving forces to use in kinetic simulations.

,...),,...,,,( 2121 ξξxxTPGG mm =

4. Thermodynamic models


Gibbs energy per mole for a solution phase is normally divided in:

phmm

Eidealmmm GGGGG ∆++∆+=

reference surface

configurational contribution(random mixing)

physical contribution

excess term

Modeling of solutions

chemical)(magnetic/ ordering e.g.

solutionregular e.g.

=∆

==∆= ∑∑==

phm

mE

C

iii

idealm

C

iiim

G

GxxRTGGxG ,ln,11


mG

(1 )B A B Bx G x G− +

AG

BG

Bx

B BG μ=A AG μ=

phmm

Eidealm GGG ∆++∆


Regular-solution type of models

etcsolutionregular -sub-Sub

solutionregular -Sub solutionRegular polynomialKister -Redlich

where

:2:1:0

...)()()( 2211

===

+−+−+=−+=

=

∑

∑∑>

kkk

LxxLxxLLxxLL

LxxG

ijjiijjiijijk

k

kjiijij

ji jijjim

E


ABL0

ABL1

ABL2 ABL3

:mEG to

onsContributi


Phases with sublattices

),)(,(),...)(,,(

),(),,(),)(,(

.....).....(.......)(...)(

1111

623

2121 21

−−++ FClKNaVaCSiMnFe

NCMoFeCrNOMnNi

BBAAtaaa

e.g. Salts, e.g. als,Interstiti

e.g. nitrides,-Carbo e.g. Oxides,:exampleFor


/

/

tj

t t tj k j

j

m

y

y N N

NG G N

=

==

∑

The site fraction is the fraction of lattice sites onsublattice t that are occupied by component k.

number of formula units

One can usually not calculate the chemical potential

/ kG N∂ ∂of a component because the derivative

cannot be taken without violating the constraints.


623

321

212121

321

321

)()()(

....),(....),(....),(

CCr

IIIII

yGy

yGG

CCBBAA

aaa

t ktk

mtk

ttI

mmI

aaa

t

eg ...

:compound alhypothetic a is where

...general, In

=

∂∂

−∂∂

+= ∑∑∑µ


The compound energy formalism

( ) ( )

vanish.energy excess and entropy mixing reference the in :Note

energies? excess and reference the about What

sublattice each on mixture Randomunit. formula of moleper

? of energy Gibbs the represent to how :exampleFor

NNCCFeFeCrCrmixm

m

yyyyyyyyRS

GNCFeCr

lnln6lnln23/

),(),( 623

+++=−


623623623623

623623

623623

)()()()()()()()(

NFemNFe

CFemCFe

NCrmNCr

CCrmCCr

refm GyyGyyGyyGyyG

NFeCFeNCrCCr

+++=

:compounds 4

The reference energy ”surface”


[ ]

)

...ln6ln23

623623623623

623

623

623 21

NFem

CCrm

CFem

NCrmFeCCrN

CCrFeCCrNNFeCCr

mCCr

t ktk

mtk

C

m

Cr

mmCCr

GGGGG

yyRTGyyG

yGy

yG

yGG

−−+=∆

+++∆+=

∂∂

−∂∂

+∂∂

+=

+

+

∑∑

µ

µ

The chemical potentials


1. Introduction – A shift in Paradigm!2. General idea behind CALPHAD3. Thermodynamic basis4. Thermodynamic models5. Some Examples 6. Extension of CALPHAD beyond thermochemistry7. Conclusive remarks


From: Saunders & Miedownik: ”Calphad -a comprehensive review”

deviation Average%4<d

5. Some examples


• FeBase• Cr23 – 27%• Ni 6 – 8%• Mo 3 – 5%• N 0.25 – 0.29%• C 0 – 0.03%

Nominal composition (SAF 2507):Fe – 25% Cr – 7% Ni – 4% Mo – 0.27% N – 0.02% C

Predict the temperature when sigma-phase becomes stable within some composition variation:


0

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

NPM

(*)

10-24

10-21

10-18

10-15

10-12

10-9

10-6

.001 1

FUNCTION PO2

2

2:PO2,NPM(SPINEL)

2

3

3:PO2,NPM(FCC_A1)

1

1:PO2,NPM(HALITE)

2

2

4

4:PO2,NPM(CORUNDUM_M2O3)

2

4

Ni-2 mass% Al at 1200 °C (TCFE7)

2 OP

Mol

e fr

actio

n ox

ides

NiAl2O4+NiO

NiAl2O4 NiO

Al2O3

Oxidation of Ni-base alloy


The data challenge (ex. Ni-base alloys)Al B C Co Cr Fe Hf Mo N Nb Ni Pd Pt Re Si Ta Ti V W

B xC x xCo x x xCr x x x xFe x x x x xHf x x x x x xMo x x x x x x xN x x x x x xNb x x x x x x x x xNi x x x x x x x x x xPd x x x x x x x x x xPt x x x x x x x x x xRe x x x x x x x x x x x xSi x x x x x x x x x x x x x xTa x x x x x x x x x x x x x x xTi x x x x x x x x x x x x x x x xV x x x x x x x x x x x x x x x x xW x x x x x x x x x x x x x x x x x xZr x x x x x x x x x x x x x x x x x x x

20 elements: 190 binary systems 184 of the 190 binary systems assessed for full range composition All Ni containing ternaries plus other ternary systems also

assessed to full range of composition (184 in total) 292 intermetallic and solution phases




6. Extension of CALPHAD beyond thermochemistry• Diffusion and phase transformations: driving forces

from CALPHAD thermodynamics• Other properties

- Diffusion coefficients- Interfacial mobilities- Interfacial energies- Stacking fault energy- Elastic constants (anisotropic behaviour)- Optical - Electronic- Magnetic- Strength, ductility etc...


Thermodynamics: Gibbs energy

Diffusion: Mobility

Phase Field Method

Langer-Schwartz

First Principles Calculation

From atoms to microstructure: The aim is to predict microstructure evolution and materials properties.

CALPHAD f(ρ)

ρ

Interfacial energy & Volume & Elastic constants


Genomic database for diffusion

• Multicomponent systems: many diffusion coefficients!• Various type of coupling effects may make it more

complicated than Fick’s law.• A CALPHAD-type of approach was suggested for

information on diffusion kinetics (Andersson-Ågren 1992)- Allowed systematic representatation of the kinetic

behaviour of multicomponent alloy systems.• DICTRA was developed in the 1990s for numerical

solution of multicomponent diffusion problems in simple geometries.


Diffusion in Ni-base alloys: Campbell et al. 2002

Simple substitutionalmodel


Growth of oxides controlled by diffusion

Atmosphere with O2

Metal α Oxide β

Me

OxMe+yO->MexOy

xMe+yO->MexOy

- Oxygen diffusion in the oxide layer gives inward growth- Metal diffusion in the oxide layer gives outward growth- Internal oxidation needs oxygen diffusion into the metal, i.e.

Oxygen diffusion through the oxide scale and the metal


Defect based models of diffusion in oxides

Vacancy mechanism operative on different sublattices. The defect structure, the vcancy content, calculated from CALPHAD databases, and the mobility parameters are stored in mobility databases.

Generalization the Wagner model!

Account for type A grain-boundary diffusion.


Experimental data on Fe tracer diffusion in spinel -Optimization of Fe mobilities

(Hallström et al. 2011)


Jonsson et al Mat Sci Forum 595-598(2008)1005

0 5 10 15 20Distance from surface [µm]M

axim

um v

olum

e fr

actio

n po

rosi

ty

Simulation of oxidation of iron using the homogenization model in DICTRA


Mechanical properties• Solution hardening (use the ”compound energy

formalism”):

( ) ( )

!parameters adjustable as taken are they Hereconstants. elastic andparameter lattice in mismatch ofncombinatio a represent parameters the and

models classical In

stress Yield :unit Formula

Amn

AyyyAyyy

yy

VaNCMM

iki

mk

kiijk

ijk

nji

k

SSHy

SSHyyij

ijjiy

b

3/2

),,...)((

'''''''''

'''

21

==

+=∆

∆+=

∑∑∑∑

∑

σ

σσσ




7. Conclusive remarks

• Models in CALPHAD can have any level of sophistication and account for:- Crystallographic structure- Lattice vibrations- Chemical order and disorder- Other phenomena such as magnetic order disorder...

• Different phases in the same material can be represented by different models and require different type of parameters.

• Quantities which are experimentally unknown, uncertain or show a large scatter may be calcaluated by ab-initio methods.

CALPHAD is the most efficient method to organise our experimental and theoretical knowledge on thermodynamics and phase equilibria.

Documents

The CALPHAD method – basis and applications