Modelling of the motion of phase interfaces coupling of thermodynamics and kinetics
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Modelling of the motion of phase interfaces; coupling of thermodynamics and kinetics. John Ågren Dept of Materials Science and Engineering Royal Institute of Technology S-100 44 Stockholm Ackowledgement: Joakim Odqvist, Henrik Larsson, Klara Asp Annika Borgenstam, Lars Höglund, Mats Hillert.

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Modelling of the motion of phase interfaces coupling of thermodynamics and kinetics l.jpg
Modelling of the motion of phase interfaces; coupling of thermodynamics and kinetics

John ÅgrenDept of Materials Science and EngineeringRoyal Institute of TechnologyS-100 44 Stockholm

Ackowledgement:

Joakim Odqvist, Henrik Larsson, Klara Asp

Annika Borgenstam, Lars Höglund, Mats Hillert


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Content thermodynamics and kinetics

  • Sharp interface

  • Finite thickness interface

    • solute drag theories

    • Larsson-Hillert

  • Diffuse interface

    • phase field

  • No interface

  • Conclusions


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Modeling of the local state of phase interface thermodynamics and kinetics (Hillert 1999)

Sharp interface

- no thickness

Finite interface

thickness – continuous

variation in properties


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Diffuse interface: no sharp boundary between thermodynamics and kinetics

interface and bulk (phase-field method)

No interface: Only bulk properties

are used, i.e. not even an operating interfacial

tieline calculated.


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Sharp interface thermodynamics and kinetics

(Stefan Problem)

The rate of isobarothermal diffusional phase transformations in an N component system may be predicted by solving a set of

N-1 diffusion equations in each phase : 2(N-1)

Boundary conditions at interface:

  • - Composition on each side 2(N-1)

  • - Velocity of interface +1

    2N-1

    These boundary conditions must obey:

    Flux balance equations N-1

    Net number of extra conditions needed

    at phase interface: N


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How to formulate the thermodynamics and kineticsN extra conditions?

Baker and Cahn (1971): N response functions. For example in a binary system   :

Simplest case: Local equilibrium.

- The interfacial properties do not enter into the problem except for the effect of interfacial energy of a curved interface (Gibbs-Thomson) and the interface velocity does not enter.


Dissipation of driving force at phase interface l.jpg
Dissipation of driving force at phase interface thermodynamics and kinetics

The driving force across the interface is consumed by two independent processes:

  • Transformation of crystalline lattice

  • Change in composition by trans-interface diffusion

    The processes are assumed independent and thus each needs a positive driving force.


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Carbon diffuses across thermodynamics and kinetics

interface from  to .

Driving force for trans-interface

diffusion:

Driving force for change of

crystalline lattice:

All quantities expressed

per mole of Fe atoms.


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Trans-interface diffusion thermodynamics and kinetics(Hillert 1960)

Combination with finite interface mobility yields:

C and Fe are functions of the composition

on each side of the interface and may be described

by suitable thermodynamic models of the  and 

phase, respectively.


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Trans-interface diffusion substitutional system thermodynamics and kinetics

Combination with finite interface mobility yields:

A and B are functions of the composition

on each side of the interface and may be described

by suitable thermodynamic models of the  and 

phase, respectively.


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For a given interface thermodynamics and kinetics

velocity the equations

may be solved to

yield the composition

on each side of interface.

x

x

x


Sharp interface with with representative composition gren 1989 l.jpg
”Sharp” interface with with representative composition thermodynamics and kinetics(Ågren 1989)


Aziz model 1982 l.jpg
Aziz model (1982) thermodynamics and kinetics

Similar as the previous sharp interface models but:


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Finite interface thickness thermodynamics and kinetics

Diffusion inside the interface  solute drag

Property

I

II

Solute drag theory (Cahn, Hillert and

Sundman etc)

III

Distance y


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Driving force = Dissipation: al. 2003)

The compositions on each side

of the phase interface depend

on interface velocity and they app-

roach each other.

Above a critical velocity trans-

formation turns partitionless.


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Driving diffusion in al. 2003)

Alloy 1

Non-parabolic growth in early

stages.

Maximum possible growth rate for alloy 1


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partitionless growth possible al. 2003)

Alloy 2

partitionless growth with spike

in  possible

Maximum possible (partitionless) growth rate

for alloy 2


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Calculated limit of the massive al. 2003)

transformation in Fe-Ni alloys

(dashed lines) calculated for

two different assumptions on

properties of interface.

(Odqvist et al. 2002)

Exp: Borgenstam and Hillert 2000


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Fe-Ni-C al. 2003)

The interfacial tieline depends on the growth rate.

(Odqvist et al. 2002)

- At high growth rates the state

is close to paraequilibrium.

- At slower rates there is a gradual

change towards NPLE.

- For each alloy composition there is

a maximum size which can be

reached under non-partitioning

conditions.

This size may be reached before

there is carbon impingement.

See also Srolovitz 2002.

Decreasing growth rate


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Ferrite formation under ”practical” conditions al. 2003)

in Fe-Ni-C (Oi et al. 2000)

Thickness of M spike in :

DM / v.

Local equilibrium, i.e. quasi paraequilibrium impossible if DM / v < atomic dimensions.

v = 4.8 10-7 ms-1

DM / v = 3.510-14 m


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Simulations Fe-Ni-C al. 2003)Odqvist et al. 2002

700 oC, uNi=0.0228


Larsson hillert 2005 l.jpg
Larsson-Hillert (2005) al. 2003)

Lattice-fixed frame

of reference

Absolute reaction

rate theory of vacancy

diffusion:

less ideal entropy of mixing



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Simulation 2a vacancy diffusion al. 2003)

with composition gradient

nucleates here

Initial composition

distance


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Solute trapping impossible! al. 2003)

Add term for cooperative mechanism


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Simulation 2b al. 2003)

vacancy + cooperative mechanism

nucleates here

distance


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Simulation 2c - other direction al. 2003)

vacancy + cooperative mechanism

nucleates here

distance


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Simulation 3 al. 2003)

Ternary system A-B-C, C is interstitial


Slide33 l.jpg

Simulation 3 al. 2003)

nucleates here


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Diffuse interfaces – phase-field method al. 2003)

Thermodynamic and kinetic properties of diffuse interfaces are needed, e.g.


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Phase-field modelling of al. 2003)

solute drag in grain-boundary migration

(Asp and Ågren 2006)


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Example 1 – grain boundary segregation al. 2003)

Equilibrium solute segregation

Equilibrium solute segregation by Cahn


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Example 1 – grain boundary segregation al. 2003)

Two grains devided by a planar grain boundary

  • Concentration distribution at t1=30 sec and t2=1 h

t1

t2


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Example 1 – grain boundary segregation al. 2003)

  • Concentration profile at t1=30 sec and t2=1 h

t1

t2



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Example 2 – dynamic solute drag al. 2003)

  • Initial state: A circular grain with =1 surrounded by another grain where  =0. Homogenous concentration of xB=0.01.

Phase-field at t=0


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Example 2 – dynamic solute drag al. 2003)

  • Concentration profile as a function of time


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Example 2 – dynamic solute drag al. 2003)

  • Velocity as a function of curvature

  • Concentration xB in the grain boundary


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Example 2 – dynamic solute drag al. 2003)

  • Smaller initial grain

  • Concentration profile as a function of time


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Example 2 – dynamic solute drag al. 2003)

  • Smaller initial grain


Phase field method contains similar steps as in sharp and finite interface modelling l.jpg
Phase-field method contains similar steps as in sharp and finite interface modelling:

  • Some properties of interface modelled.

  • Solution of a diffusion equation to obtain concentreation profile.

  • Cahn-Allen equation plays a similar role as the equation for interfacial friction.

    But: Thickness of interface must be treated not only as a numerical parameter but as a physical quatity.


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Example: finite interface modelling:Formation of WS-ferrite in steels

Loginova et al. 2004


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finite interface modelling:

Loginova et al. 2004


No interface at all l.jpg
No Interface at all finite interface modelling:

Svoboda et al. 2004 (by Onsager extremum principle)

Ågren et al. 1997 (from other principles)


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Conclusions l.jpg

Conclusions finite interface modelling:

Conclusions

Conclusions

  • Sharp interface methods are computationally simple but may show problems with convergency.

  • Solute drag models may be better than sharp interface models but have similar convergency problems.

  • Larsson-Hillert method very promising

  • Phase-field approach very powerful – no convergency problems – heavy computations.

  • No-interface methods – quick calculations, accuracy remains to be investigated.