Modelling the effect of the stress field of dislocations on ... - alemi.ca

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Modelling the effect of the stress field of dislocations on ... - alemi.ca

Effect ong>ofong> ong>theong>ong>stressong> ong>fieldong> ong>ofong>ong>dislocationsong>on carbondiffusionContextAKMC-MDCouplingComputationalmodelResultsConclusionong>Modellingong> ong>theong> ong>effectong> ong>ofong> ong>theong> ong>stressong> ong>fieldong> ong>ofong>ong>dislocationsong> on carbon diffusion in α-ironCoupling Molecular Dynamcs and Atomic KineticMonte-CarloR. G. A. Veiga, M. Perez, C. S. Becquart, E. Clouet andC. DomainAlEMI - June 7-8th 2011- Vancouver


Context and challengesEffect ong>ofong> ong>theong>ong>stressong> ong>fieldong> ong>ofong>ong>dislocationsong>on carbondiffusionGeneral objective : Dynamic and static agingStressContextAKMC-MDCouplingComputationalmodelResultsConclusionStrainPortevin - Le Chatelier ong>effectong> (Lüders banding)poorly understood (physical metallurgy challenge) !Short term objective : static ageingHow ong>theong> ong>stressong> ong>fieldong> ong>ofong> a dislocation affects carbon diffusion1 far from ong>theong> dislocation line (low/moderate ong>stressong>),2 and right in ong>theong> dislocation core (high ong>stressong>) ?


Outline...Effect ong>ofong> ong>theong>ong>stressong> ong>fieldong> ong>ofong>ong>dislocationsong>on carbondiffusion1 Context and challengesContextAKMC-MDCouplingComputationalmodelResultsConclusion2 Coupling Molecular Dynamics with Atomic KineticMonte-Carlo3 Computational model4 Results5 Conclusion


ong>Modellingong> phase transformationsEffect ong>ofong> ong>theong>ong>stressong> ong>fieldong> ong>ofong>ong>dislocationsong>on carbondiffusionFrom first principles to classical nucleation and growth ong>theong>oryContextAKMC-MDCouplingmmMolecular Dynamics« Coarse grained » MDClassical ong>theong>orie for Nucleation and GrowthComputationalmodelResultsµmInitial state 10 7 jumps 10 9 jumpsConclusionAb-initionmAtomic Kinetic Monte-Carlops ns µs ms s hour month century


ong>Modellingong> phase transformationsEffect ong>ofong> ong>theong>ong>stressong> ong>fieldong> ong>ofong>ong>dislocationsong>on carbondiffusionFrom first principles to classical nucleation and growth ong>theong>oryContextAKMC-MDCouplingComputationalmodelResultsMolecular DynamicsInitial state 10 7 jumps 10 9 jumpsConclusionAtomic Kinetic Monte-Carlo


Atomic Kinetic Monte-CarloEffect ong>ofong> ong>theong>ong>stressong> ong>fieldong> ong>ofong>ong>dislocationsong>on carbondiffusionContextAKMC-MDCouplingComputationalmodelResultsConclusionPrincipleTransition frequency( ) from i → j : w i−∆Eijj= w 0 expkT Residence time iw 1 w 2Γ⋅rw 3 w Zr 1 → τ R = − ∑ ln r 1j w jΓw 0 ≈ 10 14 HzChoice ong>ofong> a particular transition r 2Example : precipitation from a supersaturated solid solution¡¢¡£¤£¡£¢¥ ¦§¨©¢¡© ¦§©¢¡©


Molecular dynamicsEffect ong>ofong> ong>theong>ong>stressong> ong>fieldong> ong>ofong>ong>dislocationsong>on carbondiffusionContextAKMC-MDCouplingPrincipleNewton’s law m idvdt = f iPair interaction : f i = ∑ j f ij = ∑ j gradV ijThermodynamic ensembles : NVE, NVT, NPT...ComputationalmodelExample : Fusion and glass transitionResultsConclusion1⎡⎛ σ ⎞ ⎛ σ ⎞ ⎤= 4ε⎢⎜ ⎟ − ⎜ ⎟ ⎥¢ ¡£¤ £¤ r rV£¤2524V*(r)0Box Size (σ)2322T gT f21-10 1 2 3 4r*200 0.2 0.4 0.6 0.8 1 1.2 1.4kT (ε)


Taking out ong>theong> best ong>ofong> two techniquesEffect ong>ofong> ong>theong>ong>stressong> ong>fieldong> ong>ofong>ong>dislocationsong>on carbondiffusionContextAKMC-MDCouplingComputationalmodelResultsConclusionAtomic Kinetic Monte-CarloLarge timescaleFixed lattice, no ong>stressong>How to couple both methods ?Molecular DynamicsShort timescale (1 ps !)Free lattice, StressKinetic Monte-Carlo → kinetics (needs for ∆E ij )Molecular Statics → ∆E ijMolecular Dynamics → Deformation, ong>stressong>Applicationsuniform ong>stressong> (Snoek relaxation)non-uniform ong>stressong> (dislocation)non-uniform ong>stressong>, lattice change (precipitation)


Taking out ong>theong> best ong>ofong> two techniquesEffect ong>ofong> ong>theong>ong>stressong> ong>fieldong> ong>ofong>ong>dislocationsong>on carbondiffusionContextAKMC-MDCouplingComputationalmodelResultsConclusionAtomic Kinetic Monte-CarloLarge timescaleFixed lattice, no ong>stressong>How to couple both methods ?Molecular DynamicsShort timescale (1 ps !)Free lattice, StressKinetic Monte-Carlo → kinetics (needs for ∆E ij )Molecular Statics → ∆E ijMolecular Dynamics → Deformation, ong>stressong>Applicationsuniform ong>stressong> (Snoek relaxation) [Comp. Mat. Sc. 43 (2008)]non-uniform ong>stressong> (dislocation) [Phys. Rev. B 82 (2010)]non-uniform ong>stressong>, lattice change (precipitation)


The Molecular Dynamics/Statics song>ofong>tware...Effect ong>ofong> ong>theong>ong>stressong> ong>fieldong> ong>ofong>ong>dislocationsong>on carbondiffusionContextAKMC-MDCouplingComputationalmodelResultsConclusion


Computational modelEffect ong>ofong> ong>theong>ong>stressong> ong>fieldong> ong>ofong>ong>dislocationsong>on carbondiffusionContextAKMC-MDCouplingComputationalmodelResultsMolecular statics simulation boxCylinder (radius : 150 Å)Dislocations → anisotropic elasticity ong>theong>oryOuter shell (atoms kept fixed)PBC along dislocation line onlyOctahedral and tetrahedral sites mappedPositions in ong>theong> core not included !Conclusion


An Fe-C potential [Comp. Mat. Sc. 40 (2007)]Effect ong>ofong> ong>theong>ong>stressong> ong>fieldong> ong>ofong>ong>dislocationsong>on carbondiffusionContextAKMC-MDCouplingComputationalmodelResultsConclusionEam (embedded atom method) potentialPair interaction AND electronic densityE tot = 1 ∑ ∑Φ ij (r ij ) + ∑ 2iij≠iF i⎛⎝ ∑ j≠i⎞ρ(r ij ) ⎠Validated on two configurations : C in octa and tetra siteValidation ong>ofong> ong>theong> potential : from solid solution to carbides292725WellerSimulation- ln(D)23211917150.00085 0.00095 0.00105 0.001151/T


Binding energiesEffect ong>ofong> ong>theong>ong>stressong> ong>fieldong> ong>ofong>ong>dislocationsong>on carbondiffusionExample ong>ofong> ong>theong> edge dislocationContextAKMC-MDCouplingComputationalmodelResultsConclusionE b > 0 → attractionRight in ong>theong> core E b = 0.66 eV


Migration energiesEffect ong>ofong> ong>theong>ong>stressong> ong>fieldong> ong>ofong>ong>dislocationsong>on carbondiffusionContextAKMC-MDCouplingComputationalmodelMinimum energy pathTransition : octa (energy minimum) → tetra (saddlepoint) → octa (energy minimum)Carbon @ octa → full energy minimizationOong>theong>rwise → carbon to relax only on ong>theong> plane ⊥migration coordinate (octa–to–octa line)ResultsConclusion


Migration energiesEffect ong>ofong> ong>theong>ong>stressong> ong>fieldong> ong>ofong>ong>dislocationsong>on carbondiffusionContextAKMC-MDCouplingComputationalmodelMinimum energy pathTransition : octa (energy minimum) → tetra (saddlepoint) → octa (energy minimum)Carbon @ octa → full energy minimizationOong>theong>rwise → carbon to relax only on ong>theong> plane ⊥migration coordinate (octa–to–octa line)ResultsConclusionΔE ij


Migration energiesEffect ong>ofong> ong>theong>ong>stressong> ong>fieldong> ong>ofong>ong>dislocationsong>on carbondiffusionEdge dislocationContextAKMC-MDCouplingComputationalmodelResultsConclusion


Migration energiesEffect ong>ofong> ong>theong>ong>stressong> ong>fieldong> ong>ofong>ong>dislocationsong>on carbondiffusionScrew dislocationContextAKMC-MDCouplingComputationalmodelResultsConclusion


AKMC simulationsEffect ong>ofong> ong>theong>ong>stressong> ong>fieldong> ong>ofong>ong>dislocationsong>on carbondiffusionContextAKMC-MDCouplingComputationalmodelResultsConclusionFrom migration energies to overall kineticsAKMC → rigid lattice → o-sites connected by t-sitesCarbon diffusion investigated within a radius ong>ofong>10 nm around ong>theong> dislocation lineTemperatures in ong>theong> 300–600 K range200,000 trajectories per temperatureStatistics → carbon jumps ≥ 1,000Randomly chosen starting point


AKMC simulationsEffect ong>ofong> ong>theong>ong>stressong> ong>fieldong> ong>ofong>ong>dislocationsong>on carbondiffusionBoundary conditionsTwo end points → ong>theong> “core” or ong>theong> “boundary”ContextAKMC-MDCouplingComputationalmodelResultsConclusion


Effect ong>ofong> ong>theong>ong>stressong> ong>fieldong> ong>ofong>ong>dislocationsong>on carbondiffusionpossible ong>theong> plot ong>ofong> ong>theong> kinetics at a given temperature.The results are shown in Figure 3. For T 120 °C, ong>theong> evolution ong>ofong> TEP shows sigmoïdal kineticswhich have ong>theong> same amplitude and are characteristic ong>ofong> a ong>theong>rmally activated phenomenon. It isimportant to note that sigmoïdal evolutions are associated with an increase in ong>theong> Vickers microhardnesswhich reaches 30 Hv. For T 120 °C, ong>theong> sigmoïdal evolution is followed by a TEP increaseResultswhich seems to be linear according to log(t). No evolution ong>ofong> ong>theong> micro-hardness is detected during thisstage. This observation allows us to assume that ong>theong>re are two different stages during ong>theong> ageing ong>ofong>strained ULC steel.For T 170 °C, only ong>theong> last TEP increase appears, ong>theong> first stage is already finished, and a slightdecrease ong>ofong> hardness can be detected for long time.Segregation kinetics[Lavaire et al, Scripta Mater 44 (2001)] ContextAKMC-MDCouplingComputationalmodelResultsConclusionFigure 3. Strain ageing kinetics ong>ofong> ULC steel after cold rolling (50% ong>ofong> reduction) for different temperatures.


Bias on carbon diffusionEffect ong>ofong> ong>theong>ong>stressong> ong>fieldong> ong>ofong>ong>dislocationsong>on carbondiffusionContextAKMC-MDCouplingComputationalmodelResultsConclusionMean displacement vector∑−→ N −→ ɛ = P i→j δ i→jj=1where( )−∆Ei→jP i→j = exp→ transition probabilityk B Tδ i→j → jump vector from i to j⃗ɛ = ⃗0 → simple random walkFor all transitions :Same probabilitySame jump distance⃗ɛ ≠ ⃗0 → biased random walk


Bias on carbon diffusionEffect ong>ofong> ong>theong>ong>stressong> ong>fieldong> ong>ofong>ong>dislocationsong>on carbondiffusion|⃗ɛ| ≥ 0.01 Å, for T=300 KContextAKMC-MDCouplingComputationalmodelResultsConclusion


Comparison with elasticity ong>theong>oryEffect ong>ofong> ong>theong>ong>stressong> ong>fieldong> ong>ofong>ong>dislocationsong>on carbondiffusionContextAKMC-MDCouplingComputationalmodelWhat is elasticity ong>theong>ory ?BCC → Anisotropic !Carbon : “elastic dipole”Associated force moment tensor P ijStress :σ ij = − 1 V P ijResultsConclusionDislocation : elastic deformation ɛ d ijBinding energy in octahedral sites AND tetrahedral sites :E o/tb= P o/tij· ɛ d ijEnergy barrier :∆E = E t b − E o b


Comparison with elasticity ong>theong>oryEffect ong>ofong> ong>theong>ong>stressong> ong>fieldong> ong>ofong>ong>dislocationsong>on carbondiffusionBias on carbon diffusionContextAKMC-MDCouplingComputationalmodelResultsConclusion


|E eb | (meV10Comparison with elasticity ong>theong>ory5Effect ong>ofong> ong>theong>ong>stressong> ong>fieldong> ong>ofong>ong>dislocationsong>on carbondiffusionContextEnergy barrriers20(A)00 1 2 3 4 5R (nm)(B)20AKMC-MDCouplingComputationalmodelResultsConclusion|E eb | (meV)1510|E eb | (meV)15105500 1 2 3 4 5R (nm)(B)2000 1 2 3 4 5R (nm)


Comparison with elasticity ong>theong>oryEffect ong>ofong> ong>theong>ong>stressong> ong>fieldong> ong>ofong>ong>dislocationsong>on carbondiffusionResidence time (s)ContextAKMC-MDCouplingComputationalmodelResultsConclusion


Comparison with elasticity ong>theong>oryEffect ong>ofong> ong>theong>ong>stressong> ong>fieldong> ong>ofong>ong>dislocationsong>on carbondiffusionSegregation kineticsContextAKMC-MDCouplingComputationalmodelResultsConclusion


So far...Effect ong>ofong> ong>theong>ong>stressong> ong>fieldong> ong>ofong>ong>dislocationsong>on carbondiffusionContextAKMC-MDCouplingComputationalmodelResultsConclusionSummarizing...Carbon diffusion in ong>theong> vicinity ong>ofong> a dislocation → biasedrandom walkCarbon jumps faster as it gets close to ong>theong> dislocation lineThis ong>effectong> decreases quickly with increasing temperatureAttractive ong>effectong> (compared to ong>theong> bulk case)Edge ong>effectong> more important than screw


Work in progressEffect ong>ofong> ong>theong>ong>stressong> ong>fieldong> ong>ofong>ong>dislocationsong>on carbondiffusionContextAKMC-MDCouplingMS+AKMC : carbon diffusion in ong>theong> dislocation coreMapping energy minima in ong>theong> core ong>ofong> ong>dislocationsong> (MS)Energy barriers → Nudged elastic band (MS)Carbon diffusion in ong>theong> pipe (AKMC)Computationalmodel0.9ResultsConclusionEnergy (eV)0.5BulkDisloc.core00 0.05 0.1Distance (nm)


Towards AlEMI : a few ideasEffect ong>ofong> ong>theong>ong>stressong> ong>fieldong> ong>ofong>ong>dislocationsong>on carbondiffusionContextAKMC-MDCouplingComputationalmodelResultsConclusionMS+AKMC : solute migration to α/γ interfaceFe-C-X potential from Fe-C and Fe-X ?diffusion across ong>theong> interface ?MS : towards a better understanding ong>ofong> X ∗ Xong>effectong> ong>ofong> ong>theong> interface “roughness” ?MD+MC/KMC :interface migrationInterface migration with MCSolute diffusion with KMCThanks for your attention !

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