Goal: Predict solute and defect evolution near a dislocation

System: substitutional Si in Ni Approach: •  Ab initio calculation of migration barriers •  Self-consistent mean-field method

Transport of point defects near dislocations (sinks)

Venkat Manga, Zebo Li, Thomas Garnier, Maylise Nastar, Pascal Bellon, Robert Averback, Dallas R. Trinkle

Materials Science and Engineering, Univ. Illinois, Urbana-Champaign CEA Saclay, Service de Recherches de Métallurgie Physique, France

Under irradiation: continuous transport of defect fluxes to sinks •  Coupling of defects and solutes fluxes? •  Segregation, precipitation, creep? Stresses of dislocation and applied •  Inhomogeneous driving forces •  Inhomogeneous anisotropic mobilities

Stress-induced anisotropic diffusion in alloys: Complex Si solute flow near a dislocation core in Ni

BES DE-FG02-05ER46217 & NERSC

Irradiation-induced precipitation

Barbu and Ardell, Scripta metal.,9,(1975) Rehn et al., Phys. Rev. B, 30 (1984)

Dislocations Grain boundaries

•  Undersaturated Ni-Si alloys –  Precipitation of Ni3Si precipitates induced by vacancy

flux to sinks

Composition profiles at dislocation loop in CP304 post-irradiation

•  Enrichment of Si, Ni and P at the dislocation

•  Cr and Mn are depleted •  Similar segregation

behavior to grain boundaries

Si Cr

Ni Mn

Cu

C P

Was et al. Acta Mater. 59, 1220 (2011)

Solute drag: vacancy and solute flux coupling

JV / JSi negative if solutes and vacancies move

in opposite directions

JV / JSi positive if solutes and vacancies move

in same direction

Vacancy flux

Si flux Si flux

Vacancy flux

Solute flux

(a)

Vacancy flux

(b)

vacancy-solute exchange dominant

solute drag by vacancy complex

The Onsager flux equations

Linear relation between fluxes and the driving forces (Allnatt1993) Ji = Lij

j∑ Xj

JNi = LNiNi∇µNi + LNiSi∇µSi + LNiV∇µVJSi = LSiSi∇µSi + LSiNi∇µNi + LSiV∇µV

Lij : The phenomenological coefficients; X : the driving force (for e.g. gradient of chemical potential)

(i,j= 1,2,…)

In a binary system Ni-Si

Quantities of interest from phenomenological coefficients

Diffusion coefficients

DSi =kTnSi

LSiSiDSi =kTn(LSiSicSi

−LNiSicNi

)(1+ ∂ lnγSi∂ lncSi

) In the dilute limit (cSi<<1)

vacancy-mediated diffusion

Jii∑ = 0

γsi -- activity coefficient csi -- concentration of Si

JV = LVSi∇µSi + LVNi∇µNi + LVV∇µV

LSiV = −LSiSi − L NiSi G =LNiSiLSiSi

LSiV - negative if solutes and vacancies move in opposite directions - positive if solute-drag is predominant

Allnatt: Atomic transport in solids (1993)

Solute drag

Jump type

ω0 self-diffusion jump

ω1 vacancy-exchange

ω2 impurity jump

ω3 dissociation jump

ω4 association jump

Vacancy mediated diffusion in FCC Ni

(111) plane

ω1 ω3

ω3 << ω1 : Si-Va tightly bound = Si in the same direction as vacancies ω3 ≈ ω1 : Si-Va no strong interaction = Si flows opposite to vacancies

Va

Ni

Si

ω1

ω2 ω3 ω4

ω0

mig

ration

bar

rier

(eV

)

migration path

0.1

0.2

0.3

0.4

0.5

0.25 0.5 0.75

atomic jump freq.

phonon frequency enthalpy of migration

ω =

ν iIS

3N∏

ν iTS

3N−1∏

exp(−ΔFeleTS−IS

kBT)exp(−

ΔHTS−ISmig

kBT)

•  Phonon frequencies ‘ν’ within harmonic approximation

•  Thermal electronic contribution from electronic density of states

•  Enthalpy of migration from nudged-elastic band method

Jump frequencies: harmonic transition state theory

First-Principles calculation of

LSiV(δ) = LSiV(0)+dLSiVdδ δ=0

⋅δ

LSiV(ω0,ω1,ω2...) From self-consistent mean-field(1,2)

ω j (δ) =ν*(δ) ⋅exp

−ΔEj (δ)kT

$

%&

'

() From ab initio calculations

Lij (δ) : phenomenological coefficients as a function of strain

δ

δ

δ

δ

δ

-2δ δ/2

δ/2

Volumetric strain Tetragonal strain Shear strain

1. Nastar et al. Phil. Mag. (2005), 2. Nastar et al. Phil. Mag. A (2000)

Phenomenological coefficients modified by strains

Jump type Tension (δ=0.01)

δ=0 Compression (δ=-0.01)

ω0 self-diffusion jump

Bar

rier

s [e

V] 1.04 ≈ 1.10–0.07 1.10 1.17 = 1.10+0.07

ω1 vacancy-exchange 0.99 ≈ 1.05–0.07 1.05 1.12 = 1.05+0.07

ω2 impurity jump 0.87 = 0.94–0.07 0.94 1.01 = 0.94+0.07

ω3 dissociation jump 1.20 = 1.27–0.07 1.27 1.34 = 1.27+0.07

ω4 association jump 1.10 ≈ 1.16–0.07 1.16 1.23 = 1.16+0.07

Vacancy-mediated diffusion with hydrostatic strain δ

δ

δ

•  5-frequency model for FCC •  No change in the symmetry

under volumetric strain

•  The planar cage for any <110> jump •  Strain on the cage area and its

relation to the barriers

Ni Si

<110> jumps

A

cage at the transition state

ΔEi (δ) ≈ ΔEi (0)− (7eV)δ

Strain on the cage diagonal : δ

All barriers change by the same quantity with strain:

0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.21000/T [K-1]

10-40

10-35

10-30

10-25

10-20

10-15

10-10

DSi

[m2 /s]

1666.7 1250 1000 833.3 714.3 625 555.5 500 454.5

δ = 0Allison1959Polat1985

0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.21000/T [K-1]

10-30

10-25

10-20

10-15

10-10

DSi

[m2 /s]

1666.7 1250 1000 833.3 714.3 625 555.5 500 454.5

δ= -0.01δ=0δ= 0.01

Diffusion coefficient of Si in Ni: hydrostatic strain

•  Calculations similar to the 5-frequency model •  14 frequencies when including 3rd neighbor interaction •  1st neigh. 0.1eV attraction, 3rd neigh. 0.05eV repulsion •  Most hop barriers follow kinetically-resolved activation

barrier approx.: forward-reverse average ≈ constant •  Same change in activation energies with stress for all jumps

Calculated diffusivities match the experimental data

δ

δ

δ

Mobility coefficients of Si in Ni: solute drag •  From the 14 frequency model

Small change in vacancy wind with volumetric strain

100012001400160018002000T [K]

107

108

109

1010

1011

mob

ility

[mol

/eV

Å s

]

Si-Si [Eact = 0.94eV]vac-vac [Eact = 1.1eV]

1100 1150 1200 1250 1300T [K]

−1×108

0

1×108

mob

ility

[mol

/eV

Å s

]

Si-vac

Mobility coefficients of Si in Ni: vacancy wind •  From the 5 frequency model: crossover temperature of 1200K

Small change in vacancy wind with volumetric strain

500 1000 1500 2000

T [K]

−1

−0.5

0

0.5

1

LS

iV/L

SiS

i

for T=1100-1300K

1

LSiV

@LSiV

@"volume

⇡ 68

δ

δ

-2δ

Migration barriers and jump frequencies: tetragonal strain

δ[101] ≈ −δ2

δ[110] = δ

δ[101] ≈ −δ2

(111) Plane : Tetragonal strain

12 first nearest neighbor(NN) bonds break symmetry <110>: Red (8) and Blue (4)

δ>0 – Blue bonds are longer than Red δ<0 – Blue bonds are shorter than Red

Needs 15 (44) frequencies to calculate Lij matrix

δb = δ δr = -δ/2

The degeneracy is lost due to strain: 12 ΔE0 split into : 8 ΔE0

-- and 4 ΔE0--

Ni self-diffusion jumps (ω0 –type) under tetragonal strain

Strain: +0.01 Strain: 0 Strain: -0.01

Bar

rier

(e

V)

ΔE0-- 1.24 ≈1.10+0.14

ΔE0 1.10

ΔE0-- 0.96 ≈ 1.10-0.14

ΔE0-- 1.02 ≈1.10-0.07 ΔE0

-- 1.16 ≈ 1.10+0.07

δb = δ and δr = -δ/2; with δ= ± 0.01

ω0 ω0- -

, ω0 - -

•  Blue Jumps (ΔE0) increase (decrease) for δ>0 (<0) •  Red Jumps (ΔE0) decrease (increase) for δ >0 (<0)

For a strain δ = 0.01 diagonal of the cage Contracts (by -0.01) – for Blue jump

Expands (by 0.005) – for Red jump

ΔE0 (δ) ≈ ΔE0 (0)− (14eV)δdiagonal

The degeneracy is lost due to strain: 12 ΔE0 split into : 8 ΔE2

rr and 4 ΔE0bb

Si-vacancy exchange (ω2 –type) under tetragonal strain

δb = δ and δr = -δ/2; with δ= ± 0.01

•  Blue Jumps (ΔE2) increase (decrease) for δ>0 (<0) •  Red Jumps (ΔE2) decrease (increase) for δ >0 (<0)

For a strain δ = 0.01 diagonal of the cage Contracts (by -0.01) – for Blue jump

Expands (by 0.005) – for Red jump

ΔE2 (δ) ≈ ΔE2 (0)− (14eV)δdiagonal

ω2 ω2rr

, ω2bb

Strain: +0.01 Strain: 0 Strain: -0.01

Bar

rier

(e

V)

ΔE2bb 1.07 ≈ 0.94+0.14

ΔE2

0.94

ΔE2bb 0.80 ≈ 0.94-0.14

ΔE2rr 0.86 ≈ 0.94-0.07 ΔE2

rr 1.00 ≈ 0.94+0.07

Ni Si

Si-Va swing (ω1 –type) under tetragonal strain

Strain: +0.01 Strain: 0 Strain: -0.01

Bar

rier

(eV

) ΔE1rr 1.18 ≈ 1.05+0.14

ΔE1

1.05

ΔE1rr 0.92 ≈ 1.05-0.14

ΔE1rb 0.98 ≈ 1.05-0.07 ΔE1

rb 1.11 ≈ 1.05+0.07

ΔE1br 0.98 ≈ 1.05-0.07 ΔE1

br 1.11 ≈ 1.05+0.07

ω1 ω1rb

, ω1rr

, ω1br

Ni Si

Si-Va re-orientation can happen faster along particular directions

•  Blue Jumps (ΔE2) increase (decrease) for δ>0 (<0) •  Red Jumps (ΔE2) decrease (increase) for δ >0 (<0)

δb = δ and δr = -δ/2; with δ= ± 0.01

ΔE1(δ) ≈ ΔE1(0)− (14eV)δdiagonal

Si-Va dissociation jump (ω3 –type) under tetragonal strain

δb = δ and δr = -δ/2

ω3 ω3r-

, ω3b-

, ω3b-

, ω3r-

Strain: +0.01 Strain: 0 Strain: -0.01

Bar

rier

(eV

)

ΔE3b- 1.41 ≈ 1.27+0.14

ΔE3

1.27

ΔE3b- 1.12 ≈ 1.27–0.14

ΔE3r- 1.40 ≈ 1.27+0.14 ΔE3

r- 1.11 ≈ 1.27–0.14

ΔE3r- 1.18 ≈ 1.27–0.07 ΔE3

r- 1.34 ≈ 1.27+0.07

ΔE3b- 1.18 ≈ 1.27–0.07 ΔE3

b-

1.34 ≈ 1.27 - 0.07

ΔE3(δ) ≈ ΔE3(0)− (14eV)δdiagonal

•  Blue Jumps (ΔE3) increase (decrease) for δ>0 (<0) •  Red Jumps (ΔE3) decrease (increase) for δ >0 (<0)

Si-Va association jump (ω4 –type) under tetragonal strain

Strain: +0.01 Strain: 0 Strain: -0.01

Bar

rier

(eV

)

ΔE4-b 1.29 ≈ 1.16+0.14

ΔE4

1.16

ΔE4-b 1.02 ≈ 1.16-0.14

ΔE4-r 1.29 ≈ 1.16+0.14 ΔE4

-r 1.00 ≈ 1.16-0.14

ΔE4-r 1.09 ≈ 1.16-0.07 ΔE4

-r 1.21 ≈ 1.16+0.07

ΔE4-b

1.08 ≈ 1.16-0.07 ΔE4

-b 1.22 ≈ 1.16+0.07

ω4 ω4-r

, ω4-b

, ω4-b

, ω4-r

δb = δ and δr = -δ/2

ΔE4 (δ) ≈ ΔE4 (0)− (14eV)δdiagonal

•  Blue Jumps (ΔE4) increase (decrease) for δ>0 (<0) •  Red Jumps (ΔE4) decrease (increase) for δ >0 (<0)

Blue jump Red jump

Changes in the cage-diagonal explains the changes in the migration barriers

A constant is found on all jump types dΔEdδL

L

-0.01 -0.005 0 0.005 0.01Strain on Cage Diagonal, δL

0.6

0.8

1

1.2

1.4

1.6B

arrie

r hei

ght, Δ

E (e

V)

ΔE0(δ)ΔE1(δ)ΔE2(δ)ΔE3(δ)ΔE4(δ)fit line ΔE0(δ)fit line ΔE1(δ)fit line ΔE2(δ)fit line ΔE3(δ)fit line ΔE4(δ)

dΔEdδL

≈ −14eV

ΔEj (δL) ≈ ΔEj (0) – 14eV δL

Summary: All jump barriers under tetragonal strain

The blue jumps: ΔEi (δ) ≈ ΔEi (0) + 14eV δ The red jumps: ΔEi (δ) ≈ ΔEi (0) – 7eV δ

Symmetry enforces −2:1 ratio for derivative

δ

δ

-2δ

• Microscopic Master equation

Kinetic Coupling: Self-Consistent Mean-Field Model

Vaks (1993); Nastar et al. (2000; 2005; 2007) Garnier and Nastar (2011)

dP(n, t)dt

= Wn∑ ( n→ n)P( n, t)− W

n∑ (n→ n)P(n, t)

H =12!

Vijαβ

α,β ,i≠ j∑ ni

αnjβ +

13!

Vijkαβγ

α,β ,γ ,i≠ j≠k∑ ni

αnjβnk

γ +...

• Under imposed chemical potential gradient: introduce effective interactions so as to satisfy steady state P(n,t)= P0 (n)P1(n,t);

P0 (n)= exp[β(Ω0 + µαα

∑ niα

i∑ − H )]at equilibrium

with

h(t) = 12!

υijαβ (t)

α,β ,i≠ j∑ ni

αnjβ +

13!

υijkαβγ (t)

α,β ,γ ,i≠ j≠k∑ ni

αnjβnk

γ + ⋅ ⋅ ⋅

d niα

dt= − Ji→s

α

s≠i∑ and

d niαnj

β

dt= 0 Linear system relating effective

interactions and gradient of µ’s

• The effective interactions contain the kinetic coupling terms

Kinetic Coupling: Self-Consistent Mean-Field Model (2)

becomes

Ji→ jα = −Lij

(0)α δµ jα −δµi

α( )+ Lijs(1)ασ

σ ,s∑ υ js

ασ −υisασ( )+ ...

Ji→ jα = − Lij ,ss '

αβ µs 'β −µs

β( )β ,s,s '∑

• Approach can be extended to arbitrary crystallographic structures

• Requires knowledge of atomic jump frequencies

Allows for including stress effects on kinetics:

e.g., creep, transport near dislocations

full Onsager Matrix

Anisotropy in solute drag due to tetragonal strain

Perpendicular to the elongation axis

Parallel to the elongation axis

•  LSiSi > 0 in the entire temperature range •  LSiV is positive if solute-drag is predominant

Tetragonal strain with δ= ± 0.01

δ = - 0.005 δ = 0.005

300 600 900 1200 1500T [K]

-1

-0.5

0

0.5

1

L SiV

/LSi

Si

parallel to the X-Y planeperpendicular to the X-Y plane

Ts Ts

300 600 900 1200 1500T [K]

-1

-0.5

0

0.5

1

L SiV

/LSi

Si

parallel to the X-Y planeperpendicular to the X-Y plane

Ts Ts

Strain free Ts ≈ 950 K

δ

δ

-2δ

Anisotropy in solute drag due to tetragonal strain •  LSiSi > 0 in the entire temperature range •  LSiV is positive if solute-drag is predominant

δ

δ

-2δ

−0.04 −0.02 0 0.02 0.04

tetragonal strain

1100

1200

1300

1400

1500cr

oss

ov

er t

emp

erat

ure

[K

]

parallelperpendicular

@Tcrossover

@�= 1.92 ⇥ 10

4

K

Dislocation strain field

−yy

+yy+xy

−xy

+xx+yy

−xx−yy+xy

−xy

−xx−yy−xy

+xx

−xx+yy−xy

+xx+yy+xy

−yy+xy

+xx

−xx

Dislocation strain field

"volumetric

= � b4⇡r

sin✓

�shear

=b

4⇡r3

2

cos✓ cos 2✓

"bb

= � b4⇡r

✓2 sin✓ +

3

2

sin✓ cos 2✓◆

Anisotropy in LSiV due to dislocation strain field

"volumetric

= � b4⇡r

sin✓

�shear

=b

4⇡r3

2

cos✓ cos 2✓

"bb

= � b4⇡r

✓2 sin✓ +

3

2

sin✓ cos 2✓◆

0BBBB@L

0

+ 1

3

L0v

"volumetric

+ 1

6

L0tet

"bb

� 2

3

L0tet

�shear

� 2

3

L0tet

�shear

L0

+ 1

3

L0v

"volumetric

1CCCCA

L0: unstrained L L’v: volumetric strain derivative L’tet: tetragonal strain derivative

Anisotropy in LSiV due to dislocation strain field

difference between max. and min. eigenvalue of Lij

orientation of max. and min. eigenvectors of Lij

Anisotropy in LSiV relative to average LSiV at crossover

Tcrossover

unstrained L nearly 0: ratio nearly distance independent

Anisotropy in LSiV relative to average LSiV below crossover

extremely large anisotropies near core unusual contours should affect solute distribution

Tcrossover−100K

Anisotropy in LSiV relative to average LSiV above crossover

extremely large anisotropies near core unusual contours should affect solute distribution

Tcrossover+100K

Initial flow streams at Tcrossover

vacancy

strong directionality from primary anisotropy at crossover

silicon

vacancy

solute drag into core in anisotropic pattern

silicon

Initial flow streams at Tcrossover−50K

vacancy

depletion of solute from dislocation core above crossover (solute exchange)

silicon

Initial flow streams at Tcrossover+50K

Goal: Predict solute and defect evolution near a dislocation

System: substitutional Si in Ni Approach: •  Ab initio calculation of migration barriers •  Self-consistent mean-field method

Transport of point defects near dislocations (sinks)

Venkat Manga, Zebo Li, Thomas Garnier, Maylise Nastar, Pascal Bellon, Robert Averback, Dallas R. Trinkle

Materials Science and Engineering, Univ. Illinois, Urbana-Champaign CEA Saclay, Service de Recherches de Métallurgie Physique, France

Under irradiation: continuous transport of defect fluxes to sinks •  Coupling of defects and solutes fluxes? •  Segregation, precipitation, creep? Stresses of dislocation and applied •  Inhomogeneous driving forces •  Inhomogeneous anisotropic mobilities

BES DE-FG02-05ER46217 & NERSC

Stress-induced anisotropic diffusion in alloys: Complex Si solute flow near a dislocation core in Ni