Monte Carlo simulations of diffusive phase transformations: - PowerPoint PPT Presentation

Beyond Molecular Dynamics: Long Time Atomic-Scale Simulations – Dresden, March 26 - 29, 2012 Monte Carlo simulations of diffusive phase transformations: time-scale problems F. Soisson Service de recherches de Métallurgie Physique CEA Saclay C.-C. Fu, T. Jourdan, E. Martinez (LANL), M. Nastar and O. Senninger 1 BEMOD12– Dresden, March 26 - 29, 2012

Outline Atomistic Kinetic Monte Carlo simulations (AKMC) : from an atomistic § description of diffusion mechanisms to the kinetics of phases transformations in metallic alloys The precipitation kinetic pathways depends on point defect diffusion § properties key points : the dependence of the migration barriers and the vacancy concentrations with the local configuration Ab initio calculations (thermodynamics and diffusion) Diffusion model on a rigid lattice Atomistic Kinetic Monte Carlo simulations (AKMC) Applications to the decomposition of Fe-Cr concentrated solid solutions § Comparison with experiments (3D atom probe and SANS) Contributions of ab initio calculations (and their current limitations) § 2 BEMOD12– Dresden, March 26 - 29, 2012

Rigid lattice diffusion model • A-B alloy – rigid lattice with pair interactions : εAA, εAB, εBB + εAV, εBV B Possible improvements: - many-body interactions (triangle, tetrahedron, etc.) A - composition dependent interactions (-> FeCr alloys) V Γ C - temperature dependent interactions (-> ΔSmig, ΔSfor) Γ A-V Alternative approach : interatomic potentials C • Diffusion by thermally activated point defects jumps - jump frequency : depend on the local environments � mig � Δ E = − Γ ν exp AV � � AV A k T � � b - migration barriers: broken-bond models ￥ � � mig = − = S P − ( ) n − ( ) n Δ E E ( S P ) E ( ini ) ε ε ε AV sy s s y s Ai Aj kV { i 1 44 2 4 4 j , n k , n 3 saddle-point inte ractions broken-bonds • AKMC: residence time algorithm 1 = ￥ t Simulations with 106-107 atoms, PBC and 1 vacancy -> time scale : MC Г i i 3 3 BEMOD12– Dresden, March 26 - 29, 2012 BEMOD12– Dresden, March 26 - 29, 2012

Physical time scale and point defect concentrations The kinetic pathways depends on the migration barriers but also on the point defect concentrations : simulations with a constant number of point defects (e.g. NV = 1) require a time rescaling MC C V The time correction factor is not constant = MC = t t with C N / N MC V V eq C V eq C usually evolves during the phase transformations V MC � for � C (α) E (α) = eq = − t t V with C (α) exp V • For any local environment (α) : � � MC V eq C (α) kT � � V eq C (α) eq MC C = C V also provides an estimation of during the phase V V MC C (α) transformation V • A convenient choice for phase separation : pure A or pure B z = − + for E (pure A)ε ε z V AA AV 2 • Only valid when vacancy concentration remains at equilibrium. Alternative approach : AKMC with formation and annihilation mechanisms (sources/sinks) 4 BEMOD12– Dresden, March 26 - 29, 2012

One simple example = > = for for E (A) 1.4 eV E ( ) 1.0 eV B Phase separation in an A95-B5 alloy : V V AKMC simulation with 1 vacancy T = 573 K (0.6 Tc) eq eq C (A) C (B) e q MC MC C = C V = C V V V V M C M C C (A ) C (B ) V V eq C V Strong variation of the time rescaling factor Gibbs-Thomson effect : Application: vacancy concentration in non-ideal concentrated solid solutions (Mean-Field models, M. Nastar) eq eq C with ref. A slightly > C with ref. B V V % 5 BEMOD12– Dresden, March 26 - 29, 2012

Copper precipitation in α-iron : clusters mobility = > = for for E (C u) 0.9 eV E ( F e ) 2.1 eV V V Classical theories of nucleation, growth and coarsening: emission/absorption of individual solute atoms • AKMC simulation : Small Cu clusters are mobile than monomers -> coagulation mechanisms • Measurements of the cluster diffusion coefficients • CD and EKMC simulations (T. Jourdan) - the diffusion of clusters strongly accelerates the precipitation kinetics - better agreement with experiments 6 BEMOD12– Dresden, March 26 - 29, 2012

α-α’ decomposition in Fe-Cr alloys Fe-Cr alloys : a model system for ferritic-martensitic steels A special thermodynamic behavior : SANS and electrical resistivity experiments: below 10%Cr : ordering tendency V 1,2 (meV) V 1,2 (meV) above 10%Cr : unmixing tendency (Mirebeau et al, 1984) SRO SRO Ab initio calculations: connected to magnetic properties Strong vibrational entropy (exp. + ab initio ) Cr concentration Cr concentration Phase diagram “old Calphad”: no Cr solubility at low T  new phase diagram takes into account DFT calculations and experiments at low T (Bonny et al, 2010) 7 BEMOD12– Dresden, March 26 - 29, 2012

Thermodynamics: pair interaction model (M. Levesque et al. 2010) z ( ) ￥ = − = ε + ε − ε n n n • Constant pair interactions : Δ E x (1 x )Ω with Ω n 2 mix AA BB AB 2  symmetrical mixing energies and phase diagram n • Composition dependent pair interactions fitted on DFT calculations (1st and 2nd nearest-neighbor) -> asymmetrical phase diagram • Temperature dependence: εFeCr(x,T) = εFeCr(x,0) (1-T/θ) (magnetic effects, vibrational entropy) ￥ = − − Phase Diagram n n Δ E x (1 x ) L (1 x ) mix n ε ( ) x F eC r T (K) ε ( , ) x T F eC r spinodal • Alternative approach : Classical and Magnetic Cluster Expansion (Lavrentiev et al) 8 8 BEMOD12– Dresden, March 26 - 29, 2012 BEMOD12– Dresden, March 26 - 29, 2012

Diffusion properties: migration barriers • Vacancy migration barriers : vacancies-atom and saddle-point (SP) pair interactions fitted on DFT calculations of vacancy formation energies and migration barriers (DFT- SIESTA) attempt frequencies νFe and νCr fitted on ab initio , or the experimental pre- Fe Cr exponential factors for E • Self –diffusion in pure metals A = 2 2.18 1.91 (Spin-wave) V D a C A f ( )Γ A * V 0 0 mig E V 0.69 1.25 (AFM configuration) Q 2.87 3.16 exp: 2.91 exp: 3.2-3.6 (at low T) • Diffusion in dilute alloys : � Γ (10-frequency model) A = 2 D a C A ( ) 4 fΓ B * V 2 2 ﾢ Γ 3 ● Migration barriers are computed at 0K, in magnetic configurations extrapolation above the transitions temperature ? 9 BEMOD12– Dresden, March 26 - 29, 2012

Diffusion coefficients − = 13 1 Pre-exponential factor : ν 10 s D = for S ( F e ) 4.1 k (DF T, Lucas & S chaüblin, 2009) V B = mig S ( F e ) 2.1 k (Athènes and Marinica, 2010) V B  Good agreement with the experimental D0 (in iron) in iron in chromium 10 BEMOD12– Dresden, March 26 - 29, 2012

AKMC: α-α’ decomposition during thermal ageing Fe-20%Cr T = 500°C AKMC (E. Martinez, O. Senninger, CEA) 3D atom probe (Novy et al, GPM Rouen, 2009) 11 BEMOD12– Dresden, March 26 - 29, 2012

AKMC vs SANS experiments ( E. Martinez, O. Senninger, 2011) Small Angle Neutrons Scattering experiments : Bley 1994, Furusaka et al 1986 Fe – 20, 35 and 50% Cr, 500°C: good agreement with SANS experiments Fe – 40%Cr, 540°C : AKMC much slower than SANS experiments The Curie temperature decreases with the Cr content: effect of the ferro-paramagnetic transition? 12 BEMOD12– Dresden, March 26 - 29, 2012

Effect of the magnetic transition on the kinetics O. Senninger, E. Martinez et al. 2012 Evolution of the Curie temperature with the composition Decrease of the migration barriers at TC (fitted on the experimental diffusion coefficients) 13 BEMOD12– Dresden, March 26 - 29, 2012

Radiation induced segregation and precipitation + < Undersaturatedsolidsolution withan unmixingtendency (ε ε - 2ε 0) AA BB AB = − = eq C 5% C 8% B B v v � i i � D / D 0.06 and D / D 1 B A B A = − − = 6 1 G 10 dpa.s T 800K Solute concentration profile v Point defect sink A B ~ 50 nm B A PD sink v 0.01 dpa 0.33 dpa 2.58 dpa 14 BEMOD12– Dresden, March 26 - 29, 2012

Conclusions AKMC simulations: a detailed description of thermodynamic and diffusion properties - dependence of point defect jump frequencies and concentrations with the local atomic environment - correlation effects (↔ theory of diffusion in alloys) But time consuming: → cluster dynamics, OKMC, EKMC Ab initio calculations : → reliable parameters at 0K temperature effects : vibrational entropy of mixing, vacancy formation and migrations entropies - in pure metals - in concentrated alloys ? – Fe-Cr alloys: importance of magnetic transitions Perspectives : radiation induced segregation, spinodal decomposition in Fe-Cr alloys 15 BEMOD12– Dresden, March 26 - 29, 2012