Workshop « bridging space-time scale… », DRESDEN, September 2016

Maylise NastarCEA Saclay, France

Bridging space-time scales in the field of diffusion-controlled

phase transformations

Part of this work was supported by

o the joint program "CPR ODISSEE" funded by AREVA, CEA, CNRS, EDF and Mécachromeunder contract n°070551.

o NEEDS Materiauxo JPNM/Matisse/Mefisto

Atomic diffusion model

Macroscopic diffusion and flux coupling: diffusion equations

d ≈ aΔt ≈ a2 max(ωVA)

d ≈ naΔt ≈ d 2 DB

nano2

(10 )100 V VA

d n at n C»

D » w

Nucleation in dilute alloysCluster Dynamics, OKMC

A multiscale approach for diffusion and phase transformations

d ≈ Vat3 /CBΔt ≈ d 2 max(Mcluster )

Nanoscale diffusion & spinodal decompositionin concentrated alloys: phase field methods

𝑤"#=Γ"𝑒& '()

Growth and coarsening

d ≈ Vat3 /CBΔt ≈ d 2 max(Mcluster )

JB = − Lijj∑ ∇µ j

Atomic diffusion model

Macroscopic diffusion and flux coupling: diffusion equations

d ≈ aΔt ≈ a2 max(ωVA)

d ≈ naΔt ≈ d 2 DB

nano2

(10 )100 V VA

d n at n C»

D » w

Nucleation in dilute alloysCluster Dynamics, OKMC

A multiscale approach for diffusion and phase transformations

d ≈ Vat3 /CBΔt ≈ d 2 max(Mcluster )

Nanoscale diffusion & spinodal decompositionin concentrated alloys: phase field methods

𝑤"#=Γ"𝑒& '()

Growth and coarsening

d ≈ Vat3 /CBΔt ≈ d 2 max(Mcluster )

Interactions between cells?nanoscale fluctuations and nucleation?

Flux couplings?

JB = − Lijj∑ ∇µ j

Irradiation of Fe-15 % Cr by e- 1MeV (THT)T=773 K, F=3.9x10-5 dpa/s, dose=0.2 dpaO. Tissot, E. Meslin et al. (CEA-SRMP & Univ. de Rouen)

Atom Probe Tomography(GPM,Rouen)TEM – Bright field (CEA Saclay)

Microstructure of point defectsInterstitial loops a<100>

Radiation Induced Segregation (RIS) of Cr at loops due to flux couplings

A typical phenomenon involving bothPoint defect-solute cluster reactions and flux coupling

what is the effect of point-defect microstructure on RIS?what is the effect of RIS on the point defect microstructure?

200 nm

Continuous diffusion equations

Negative flux coupling

Spatial cluster dynamics (Crescendo)Object kinetic Monte Carlo

A typical phenomenon involving bothcluster reactions and flux coupling

DissociationCluster mobility(positive flux coupling)

VJVBJ

VAJ

sink

JAV = −

CVirr

CVeq (LAV∇µV + LAA

V ∇µA + LABV ∇µB)

VC

Onsager: positive & negative flux couplingClouet, E. ASM Handbook 2009, 22A, 203–219 .Jourdan, T. et al. Acta Mater. 2010, 58, 3295–3302.M. Nastar & F. Soisson, Comprehensive…

Continuous diffusion equations

Positive flux coupling

Spatial cluster dynamics (Crescendo)Object kinetic Monte Carlo

A typical phenomenon involving bothcluster reactions and flux coupling

DissociationCluster mobility(positive flux coupling)

VJVBJ

VAJ

sink

JAV = −

CVirr

CVeq (LAV∇µV + LAA

V ∇µA + LABV ∇µB)

VC

Onsager: positive & negative flux couplingClouet, E. ASM Handbook 2009, 22A, 203–219 .Jourdan, T. et al. Acta Mater. 2010, 58, 3295–3302.M. Nastar & F. Soisson, Comprehensive…

Outline

Ø Principles oftheself-consistentmean field theory (SCMF)

Ø Kinetic characterization ofclustersfrom theatomic scale

Ø SCMFbased - ClusterDynamicsextended toheterogeneous systems

Ø SCMFbased - PhaseFieldEquations

Principles of the Self-Consistent Mean Field theory (SCMF)

Atomic diffusion model on a rigid lattice

Application of a driving force

A mean field theory:

diffusion of particle A within a mean field

Non-equilibrium pair variables AVComputation of the coresponding fluxesIdentification of the Lij coefficients

*Nastar et al. Phil. Mag. A 2000, 80, 155–184.

𝜔+,"#

𝛻𝜇"

𝐽"

𝜔+,#

𝛻𝜇"

𝐽"

𝜔+,"#

𝛻𝜇"

𝐽"

i

i j

j

ji

SCMF theory: introduction of a non equilibriumdistribution function

Non equilibrium distribution function

Equilibrium distribution function

niα = 1, ni

V = 0njV = 1, nj

α = 0

𝜔+,"#

𝛻𝜇"

𝐽"i j

Pair correlators versus effective interactions

niα = ni

αP(n,t)n∑

niαnj

β = niαnj

βP(n,t)n∑

P(1)(n,t) = exp δΩ + δµiαni

α − 12

vijαβ (t)ni

αnjβ

i, jα ,β

∑i,α∑

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

kBT⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

P(0)(n) = exp Ω + µα niα − H (n)

i∑

α∑⎛

⎝⎜⎞⎠⎟

kBT⎡

⎣⎢

⎤

⎦⎥

P(n,t) = P(0)(n)P(1)(n,t)

Kijαβ = ni

αnjβ − ni

α njβ

Kijαβ − Kij

βα = niαnj

β (0)vijαβ − vij

βα( )

SCMF Kinetic equations from the Master equation

Master equation

10

Identification with Onsager’s flux and computation of Lab

𝒗𝒊𝒋𝑽𝜶 solution of the pair kinetic equations

dP (n,t)dt

= W n '→ n( )P(n ')−W n→ n '( )P(n)⎡⎣ ⎤⎦n '∑

JA = − Lαββ∑ β∇(µβ − µV )

Atomic fluxes

Kinetic correlations = jump efficiency

d niB

dt= −∇i JB ⇒ JB = −z ω i,i+1

BV niBni+1

V β ∇i (µB − µV )+ vi,i+1VB − vi,i+1

BV⎡⎣ ⎤⎦

d niBni+1

V

dt= ω i, i+1

BV niBni+1

V (0)∇i (µB − µV )− ni

Bω i+1,i+2V A ni+1

V ni+2A (0)

∇i+1(µA − µV )

+q1(vi,i+1AV − vi,i+1

VA )+ q2 (vi,i+2AB − vi,i+2

VB )

KINETIC CHARACTERIZATION OF CLUSTERS FROM THE ATOMIC SCALE

*T. SCHULER and M. NASTAR, PRB 93 (22), 224101

Breakdown of the Onsager matrix for a dilute alloy

Equal because of detailed balance

Mobility without dissociation

Kinetic correlations = jump efficiency

From the SCMF theory:

Dilute alloy

*T. Schuler & M. Nastar, PRB 93 (22), 224101

Equilibrium clusterTransport coefficients

Lαβ = ci[ ]i∑ Lαβ

eq (ci )

Non equilibrium Onsager transport coefficient

Cluster reactions

The mobility of a cluster 13

Mobility without

dissociation

A consistent definition of the cluster radius (Rc) is not only

thermodynamic, but also depends on kinetic correlations

SCMF in the cluster volume (no association/dissociation jumps) :

Rc

*T. Schuler & M. Nastar, PRB 93 (22), 224101

Model I: Highest barrier modelModel IV: R=Rth=1.23 aModel V: R=3.2 a

SCMF mobility compared to the highest barrier model

Mobility of VC pairs in Fe*T. Schuler & M. Nastar, PRB 93 (22), 224101

Fe(C)

The association-dissociation term 15

Rc

Dissociation - Association

Equal because of detailed balance

There are different dissociation paths => the AD matrix does not reduce to a scalar

Systems Fe(X): Monte Carlo versus SCMF

XVM

*L. Messina, T. Schuler, P. Olsson, M. Nastar

τ = a2

LVV (XV ) AD

l = MXVτ

Monte Carlo simulations: in coll. with M. Chiapetto et al. (SCK, Belgium)

Physics of flux coupling within cluster dynamics

Breakdown of the Onsager matrix into cluster contributions for dilute systems

• Exchange Correlations between A/D and M jumps• Contains the physics of negative flux coupling

Association jump Mobility jump Dissociation jump

vacancy

soluteNegative flux couplingDue to exchange correlations

SCMF-BASED CLUSTER DYNAMICSEXTENDED TO HETEROGENEOUSSYSTEMS

*T. SCHULER, T. JOURDAN, M. NASTAR

Homogeneous Cluster Dynamics

Association coefficient GAWaite formulae

κ =2MCj

λγ

Lγγ (Ci ,Cm )ADm= j ,k∑

γ =α ,β∑

Ck

Cj

Ci

Dissociation coefficient GDGlobal detailed balance condition

ΓA Cj +Ck →Ci( ) = 4πR(Ck )2MCj

(R(Ck )+κ )

ΓD Ci →Cj +Ck( ) = 1Ωat

ΓA Cj +Ck →Ci( )exp F(Cj )+ F(Ck )− F(Ci )kBT

⎛⎝⎜

⎞⎠⎟

*Clouet, E. ASM Handbook 2009, 22A, 203–219 .*Jourdan, T. et al. Acta Mater. 2010, 58, 3295–3302.

Heterogeneous Cluster Dynamics*T. Schuler, T. Jourdan, M. Nastar

Vacancy diffusion mechanism, 3 species: monovacancy V, solute B, pair BV

Flux – part 1: migration of clusters

∂ V[ ]∂t

= −∇JVCD − ΓA V[ ] B[ ]+ ΓD VB[ ]

∂ B[ ]∂t

= −∇JBCD − ΓA V[ ] B[ ]+ ΓD VB[ ]

∂ BV[ ]∂t

= −∇JBVCD + ΓA V[ ] B[ ]− ΓD VB[ ]

⎧

⎨

⎪⎪⎪

⎩

⎪⎪⎪

JVCD = −MV∇ V[ ]

JBVCD = −MBV∇ BV[ ]

Heterogeneous Cluster Dynamics

Different cluster concentrations between two adjacent cells leads to asymmetric A-D reactions and fluxes between cells.

*T. Schuler, T. Jourdan, M. Nastar

JV

Flux – part 2: Association-Dissociation of clusters

AD fluxes obtained from TIPat local equilibrium, CD and TIP fluxes are put equalkBT

∇ B[ ]B[ ] = ∇µB

kBT∇ V[ ]V[ ] = ∇µV

kBT∇ BV[ ]BV[ ] = ∇µB +∇µV

⎧

⎨

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

JVCD + JBV

CD = JVTIP

JBCD + JBV

CD = JBTIP

⎧⎨⎪

⎩⎪

Cluster Dynamics versus Monte Carlo

0 50 100 150 200 250 300 350 400 450 5000,0

5,0x10-5

1,0x10-4

1,5x10-4

2,0x10-4

2,5x10-4

3,0x10-4

3,5x10-4

CB

plane number

Atomic Monte Carlo Cluster Dynamics Crescendo

0 50 100 150 200 250 300 350 400 450 5000,0

1,0x10-4

2,0x10-4

3,0x10-4

4,0x10-4

5,0x10-4

6,0x10-4

7,0x10-4

CB

plane number

Atomistic Monte Carlo Cluster Dynamics Crescendo

GB=GA

EBV=0.1 eV

Monte Carlo simulations N=512*256*256/4

Monte Carlo eventsRandom replacement of an atom by a vacancyatrateF*NExchange of a vacancy with a nn atom (A or B)

Boundary conditionsVacancy at sinks is removed and randomly replaced by atom A or B with the constraint of constant solute concentration.

ClusterDynamicsBoundary conditionsCB=CBV=0 at sinks

Simulation of vacancy irradiation segregationT=500 K, CB=2.10-4, F=5. 10-10 dpa.s-1

𝑤𝑖𝑡ℎ𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 ∶ 𝑤"#=Γ"𝑒&'(GHI)() andWBV

GB=10 GA

Steady state profiles of solute

perfect sink

perfect sink

SCMF-BASED PHASE FIELDEQUATIONS

*M. NASTAR, PRB 90 (14), 144101

Nanoscale interdiffusion experiments*O Senninger, F Soisson, E Martínez, M Nastar, CC Fu, Y Bréchet,Acta Materialia 103, 1-11

time

Slope= -Log(wave-amplitude)

Phenomenological Cahn-Hilliard theory

slope=2Worder (=stiffness k)

annealing

D

Multilayers of isotopic atomsWorder = 0 ⇒ D (Λ) = D (0) ?

Something is missingin the Cahn-Hilliard theory

A kinetic stiffness parameter challenging the Cahn-Hilliard phenomenological equation

*M. Nastar, PRB 90 (2014), 144101

time

Slope=Log(wave-amplitude)

annealing

Composition gradient-energy Composition gradient-correlation

Interdiffusion coefficient as a function of k

κ E :κC :

Fe-Cr, T=700 K

2Dk

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

Kapp

a (e

V)

cCr

AKMC-Dumbbell AKMC-Vacancy AKMC-Direct exchange

D = M f "+ (κ th +κ kin )k2⎡⎣ ⎤⎦

Interdiffusion in binary alloys with non uniform driving forces

Beyond the Onsager coefficients

JB = −LBB∇µBV − LAB∇µAV −QBB∇3µBV −QAB∇

3µAV

A purely correlation term

MRS 2012

Uniform driving force Non uniform driving force

Conclusion and perspectives

Atomic diffusion theory SCMF could be used to properlybridge space-time scales in diffusion controlled phenomena

-Heterogeneous object oriented simulation methods-A rigourous definition of cluster kinetic properties-Modeling of flux coupling phenomena

-New diffusion equation for the phase field method-Nanoscale diffusion experiments: a way to get detailedinformation on the alloy diffusion properties

-Perspectives: Off-lattice modeling: a network specific to each cluster