Correlation between electronic structure, magnetism and physical properties of Fe-Cr alloys: ab initio modeling Igor A. Abrikosov Department 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 stability resistance to neutron Accelerator of protons radiation induced swelling Cooling media corrosion resistance increased ductile to brittle Fuel transition temperature Target
In collaboration with: P. Olsson, EDF R&D, France A. V. Ponomareva, MIS&A, Russia A. V. Ruban, KTH, Sweden L. Vitos, KTH, Sweden J. 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 Original idea for this figure belongs to Chris Wolverton Ford Motor Company Microstructure -Grains 1 10 mm Properties High cycle fatigue Ductility Microstructure -Phases 100 500 microns Properties Yield strength Ultimate tensile strength High cycle fatigue Low cycle fatigue Thermal Growth Ductility Microstructure -Phases 3-100 nanometers Properties Yield strength Ultimate tensile strength Low cycle fatigue Ductility Atoms 10-100 Angstroms Properties Thermal Growth
F F Structures: D A A B C C B D C
F = k BT ln Z Z = s exp E kt s
A atom B atom A x B 1-x alloy n(r), the electron density
LDA, GGA, etc. VASP, Wien2k,CASTEP, ABINIT, KKR, etc. 2 +V KS (x 1,R 1,R 2,...,R N 2m I ) φ i (x 1,R 1,R 2,...,R N e I ) =ε i φ i (x 1,R 1,R 2,...,R N I ) h2 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
= c + (1-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)
F = k BT ln Z Z = s exp E kt s r σ = s { σ } i σ i = 1 if site i is occupied by atom A -1 otherwise σ σ i j s, σ σ i j σ k s,... 1 1-1 1 1 1-1 1 1-1 1 1-1 1 1-1 1 1 1-1 E tot = V ( 0 ) + V (1) σ + s V ( 2, s ) σ i σ j + s V ( 3, s ) σ i σ j σ k +...
Calculations of effective interatomic potentials The generalized perturbation method 1. Calculate electronic structure of a random alloy (for example, use the CPA): 2. E one ~ A ( B 1 ( RR') = Eone( c) + VRR' σ iσ j... -determine a perturbation 2 RR' where the effective interatomic interactions are given by an analytical formula: g, t of the band energy due to small varioations of the correlation functions ) V RR ' = 1 π de { A B A t γ t γ t } R RR ' R ' R ' R R
Calculations of effective interatomic potentials The Connolly-Williams method 1. Choose structures fcc L1 2 L1 0 DO22 with predefined [ ] (2,1) (2,2) (3,1) correlation functions σ iσ j, σ iσ j, σ iσ jσ k,... 2. Calculate E tot : E(fcc) E(L1 2 ) E(L1 0 ) E(DO22) E tot = V ( 0 ) + V (1) σ + s V ( 2, s ) 2 If N then { } are found by L.S.M : f... pot N str V E V = m m f f σ i σ j... min
The Monte Carlo method Calculations of averages at temperature T: Z T k E A A s B s s = exp Create the Marcov chain of configurations: = T k E Z P B s s exp 1 Balance at the equilibrium state: = T k E s s W T k E s s W B s B s ' )exp ' ( ')exp ( E Δ 1) (0 exp 0 0 > Δ > Δ Δ r r T k E E E B Atoms exchanged ΔE
Multiscale simulations of radiation damage in Fe-Cr alloys J. Wallenius, I. A. Abrikosov, P. Olsson et al., J. Nucl. Mater. 329-333, 1175 (2004).
Atom Crystal E F VALENCE CORE E tot occ = E atom i n i tot core = E n + i i E F E En( E) de valence
Atom Crystal E F VALENCE E BOND = E crystal tot E atom tot = E F valence ( E E atom valence ) n( E) de
Transition metal E F Zr E BOND = E crystal tot E atom tot = E F valence ( E E atom valence ) n( E) de
Transition metal Nb E F E BOND = E crystal tot E atom tot = E F valence ( E E atom valence ) n( E) de
Transition metal Mo E F E BOND = E crystal tot E atom tot = E F valence ( E E atom valence ) n( E) de
Transition metal Tc E F E BOND = E crystal tot E atom tot = E F valence ( E E atom valence ) n( E) de
Transition metal Ru W E F E BOND = E crystal tot E atom tot = E F valence ( E E atom valence ) n( E) de
Transition metal Ru W E F E BOND = 1 20 WN d (10 N d )
theory (nm) exp (fm) exp (nm)
Transition metal Ru W E F E BOND = 1 20 WN d (10 N d )
Transition metal Fe E F E BOND = 1 20 WN d (10 N d )
E F atom E = ( E E ) n( E) de BOND up spin band valence Fe E F + E F spin down band atom ( E E ) n( E) de valence 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.