Post on 16-Jan-2022
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
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