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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
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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
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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
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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
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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
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F F
C
A
BC
D
Structures:
A
B
CD
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∑ ⎟⎠⎞
⎜⎝⎛ −=−=
s
sB kT
EZTkF exp Zln
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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!
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Ordered compounds
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Solution phases: supercell method
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2X2X7, segregation energy -0.043 eV
A. V. Ponomareva et al., Phys. Rev. B 75, 245406 (2007)
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3X3X9, segregation energy 0.090 eV
A. V. Ponomareva et al., Phys. Rev. B 75, 245406 (2007)
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Solution phases: coherent potential
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=
+ (1-c)c
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bcc Cr-Fe, magnetic moments
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bcc Cr-Fe, mixing enthalpies
P. Olsson, I. A. Abrikosov, L. Vitos, and J. Wallenius, J. Nucl. Mater. 321, 84 (2003)
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P. Olsson, I. A. Abrikosov, and J. Wallenius, Phys. Rev. B 73, 104416 (2006)
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}{ 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
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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
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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
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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
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Multiscale simulations of radiation damage in Fe-Cr alloysJ. Wallenius, I. A. Abrikosov, P. Olsson et al.,
J. Nucl. Mater. 329-333, 1175 (2004).
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Atom Crystal
∑=occ
iatomitot nEE ∫∑ +=
FE
valence
core
iitot dEEEnnEE )(
CORE
VALENCE
EF
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Atom Crystal
∫ −=−=FE
valence
atomvalence
atomtot
crystaltotBOND dEEnEEEEE )()(
VALENCE
EF
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Transition metal
∫ −=−=FE
valence
atomvalence
atomtot
crystaltotBOND dEEnEEEEE )()(
EFZr
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Transition metal
∫ −=−=FE
valence
atomvalence
atomtot
crystaltotBOND dEEnEEEEE )()(
EFNb
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Transition metal
∫ −=−=FE
valence
atomvalence
atomtot
crystaltotBOND dEEnEEEEE )()(
EFMo
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Transition metal
∫ −=−=FE
valence
atomvalence
atomtot
crystaltotBOND dEEnEEEEE )()(
EFTc
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Transition metal
∫ −=−=FE
valence
atomvalence
atomtot
crystaltotBOND dEEnEEEEE )()(
EFRu W
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Transition metal
EFRu
)10(201
ddBOND NWNE −−=
W
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theory (nm)
exp (nm)exp (fm)
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Transition metal
EFRu
)10(201
ddBOND NWNE −−=
W
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Transition metal
EFFe
)10(201
ddBOND NWNE −−=
-
EFFe
∫ −=FE
bandupspin
atomvalenceBOND dEEnEEE )()(
∫ −+FE
banddownspin
atomvalence dEEnEE )()(
DIFFERENT PROPERTIES !!!
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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 :
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