Correlation between electronic structure, magnetism and physical properties of Fe-Cr alloys: ab initio

modeling

Igor A. AbrikosovDepartment of Physics, Chemistry, and Biology (IFM), Linköping University, Sweden

Fe-Cr alloys

• Are the base for many important industrial steels• Used as cladding material in fast neutron reactors• Low Cr steels, up to 10 % Cr, show:

anomalous stabilityresistance to neutron radiation induced swellingcorrosion resistance increased ductile to brittle transition temperature

Accelerator of protons

Cooling media

Fuel

Target

In collaboration with:

P. Olsson, EDF R&D, France A. V. Ponomareva, MIS&A, RussiaA. V. Ruban, KTH, SwedenL. Vitos, KTH, SwedenJ. Wallenius, KTH, Sweden

CONTENTS :

• Phase equilibria in solid solutions: theoretical background

• First-principles methods: recent developments

• Fe-Cr alloys: first-principles results• Discussion: understanding of first-principles

results• Conclusions

Microstructure – Property Relationships

Engine Block≅ 1 meter

Microstructure - Grains≅ 1 – 10 mmProperties• High cycle fatigue• Ductility

Microstructure - Phases≅ 100 – 500 micronsProperties• Yield strength• Ultimate tensile strength• High cycle fatigue• Low cycle fatigue• Thermal Growth• Ductility

Microstructure - Phases≅ 3-100 nanometersProperties• Yield strength• Ultimate tensile strength• Low cycle fatigue• DuctilityOriginal idea for this figure belongs to Chris Wolverton

Ford Motor Company

Atoms≅ 10-100 AngstromsProperties• Thermal Growth

F F

C

A

BC

D

Structures:

A

B

CD

∑ ⎟⎠⎞

⎜⎝⎛ −=−=

s

sB kT

EZTkF exp Zln

A atom B atom

AxB1-x alloy

n(r) , the electron density

−

h2

2me∇2 +VKS(x1,R1,R2,...,RNI )

⎡

⎣ ⎢

⎤

⎦ ⎥ φi(x1,R1,R2,...,RNI ) =εiφi(x1,R1,R2,...,RNI )

LDA, GGA, etc.

VASP, Wien2k,CASTEP,ABINIT, KKR, etc.

Supercell, CPA, etc.

It is NOT a trivial task to run ab initio software!

Ordered compounds

Solution phases: supercell method

2X2X7, segregation energy -0.043 eV

A. V. Ponomareva et al., Phys. Rev. B 75, 245406 (2007)

3X3X9, segregation energy 0.090 eV

A. V. Ponomareva et al., Phys. Rev. B 75, 245406 (2007)

Solution phases: coherent potential

=

+ (1-c)c

bcc Cr-Fe, magnetic moments

bcc Cr-Fe, mixing enthalpies

P. Olsson, I. A. Abrikosov, L. Vitos, and J. Wallenius, J. Nucl. Mater. 321, 84 (2003)

P. Olsson, I. A. Abrikosov, and J. Wallenius, Phys. Rev. B 73, 104416 (2006)

}{ is σσ =r

⎩⎨⎧

=otherwise

atomby occupied is site if

1- 1 Ai

iσ

1 1 -1 1 11 -1 1 1 -11 1 -1 1 1-1 1 1 1 -1

,... ,s

kji

s

ji σσσσσ

∑ ⎟⎠⎞

⎜⎝⎛ −=−=

s

sB kT

EZTkF exp Zln

...),3(),2(

)1()0(

++

++=

∑∑s

kjis

sji

s

tot

VV

VVE

σσσσσ

σ

Calculations of effective interatomic potentialsThe generalized perturbation method

1. Calculate electronic structure of a randomalloy (for example, use the CPA):

)( ,~ BAtg

...21)(

)'(

''

RR

jiRR

RRoneone VcEE σσ∑+= -determine a perturbationof the band energy due tosmall varioations of the correlation functions

2.

where the effective interatomic interactionsare given by an analytical formula:

{ }∫−= ARRRBRRRARRR tttdEV ''''1 γγπ

Calculations of effective interatomic potentials

The Connolly-Williams method

1. Choose structures fcc L12 L10 DO22

2. Calculate Etot: E(fcc) E(L12) E(L10) E(DO22)

with predefinedcorrelation functions [ ],... , , )1,3()2,2()1,2( kjijiji σσσσσσσ

...),2()1()0( ∑++=s

jis

tot VVVE σσσ

min2

...}{ :L.S.Mby found are thenIf =∑⎥⎥⎦

⎤

⎢⎢⎣

⎡∑−≤

m f fVfmEVNN strpot

The Monte Carlo method

Calculations of averages at temperature T:Z

TkEA

A s Bs

s∑ ⎟⎟⎠

⎞⎜⎜⎝

⎛−

=exp

Create the Marcov chain of configurations: ⎟⎟⎠

⎞⎜⎜⎝

⎛−=

TkE

ZP

B

ss exp

1

Balance at the equilibrium state: ⎟⎟⎠

⎞⎜⎜⎝

⎛−→=⎟⎟

⎠

⎞⎜⎜⎝

⎛−→

TkEssW

TkEssW

B

s

B

s 'exp)'(exp)'(

EΔ )10( exp 00

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

≤≤>⎥⎦

⎤⎢⎣

⎡ Δ−>Δ

≤Δ

rrTkEE

E

B

Atoms exchanged

ΔE

Multiscale simulations of radiation damage in Fe-Cr alloysJ. Wallenius, I. A. Abrikosov, P. Olsson et al.,

J. Nucl. Mater. 329-333, 1175 (2004).

Atom Crystal

∑=occ

iatomitot nEE ∫∑ +=

FE

valence

core

iitot dEEEnnEE )(

CORE

VALENCE

EF

Atom Crystal

∫ −=−=FE

valence

atomvalence

atomtot

crystaltotBOND dEEnEEEEE )()(

VALENCE

EF

Transition metal

∫ −=−=FE

valence

atomvalence

atomtot

crystaltotBOND dEEnEEEEE )()(

EFZr

Transition metal

∫ −=−=FE

valence

atomvalence

atomtot

crystaltotBOND dEEnEEEEE )()(

EFNb

Transition metal

∫ −=−=FE

valence

atomvalence

atomtot

crystaltotBOND dEEnEEEEE )()(

EFMo

Transition metal

∫ −=−=FE

valence

atomvalence

atomtot

crystaltotBOND dEEnEEEEE )()(

EFTc

Transition metal

∫ −=−=FE

valence

atomvalence

atomtot

crystaltotBOND dEEnEEEEE )()(

EFRu W

Transition metal

EFRu

)10(201

ddBOND NWNE −−=

W

theory (nm)

exp (nm)exp (fm)

Transition metal

EFRu

)10(201

ddBOND NWNE −−=

W

Transition metal

EFFe

)10(201

ddBOND NWNE −−=

EFFe

∫ −=FE

bandupspin

atomvalenceBOND dEEnEEE )()(

∫ −+FE

banddownspin

atomvalence dEEnEE )()(

DIFFERENT PROPERTIES !!!

CONCLUSIONS :

• First-principles simulations can be carried out for real materials of technological importance.

• The results allow for the cautious optimism, and the choice of methodology should depend on the problem at hand.

• The mixing enthalpy for paramagnetic alloys is positive at all concentrations, in excellent agreement with experiment.

• On the contrary, ferromagnetic bcc Fe-Cr alloys are anomalously stable at low Cr concentrations. Therefore, magnetic effects must be taken into account in simulations.

• The stability of the ferromagnetic alloys is determined by the strong concentration dependence of interatomic interactions in this system, which therefore must be taken into account in simulations.

Correlation between electronic structure, magnetism and physical properties of Fe-Cr alloys: ab initio modeling���Fe-Cr alloysIn collaboration with:CONTENTS :Microstructure – Property RelationshipsCalculations of effective interatomic potentialsCalculations of effective interatomic potentialsThe Monte Carlo method CONCLUSIONS :