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Atomistic Mechanisms forAtomistic Mechanisms for Grain Boundary MigrationGrain Boundary Migration Overview of Atomistic Overview of Atomistic
Simulations of Grain Boundary Simulations of Grain Boundary MigrationMigration
Hao Zhang 1, David J. Srolovitz 1,2
1 Princeton University2 Yeshiva University
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• U-shaped half loop geometry
*v MP M M • FCC Aluminum <111> Tilt Grain Boundary
• EAM – Al
• Periodic along X and Z
M *v Mw w
gAv
w
gA*M M
Curvature-driven Grain Boundary Curvature-driven Grain Boundary MigrationMigration
v(y)
Z
• Local Velocity
•Steady-state Velocity
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• Reduced mobility increases with increasing temperature
• Mobility shows maxima at low Σ misorientations
Reduced Mobility vs. MisorientationReduced Mobility vs. Misorientation
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Stress-Driven Boundary MigrationStress-Driven Boundary Migration• Molecular dynamics in NVT ensemble
• EAM-type (Voter-Chen) potential for Ni
• Periodic boundary conditions in x and y
• One grain boundary & two free surfaces
• Fixed biaxial strain, =xx=yy• Source of driving force is the elastic energy
difference due to crystal anisotropy
• Driving force is constant during simulation
• Linear elasticity:
• At large strains, deviations from linearity
occur,
determine driving force from the difference
of the strain energy in the 2 grains:
2xx yy 1 1
1;P
2
2
1 2
(2) (2) (1) (1) 2 31 2
1 12 3
xx yy
xx yy xx yy
A A
P d
X
Y
Z
Grain Boundary
Free Surface
Free Surface
Gra
in 2
Gra
in 1
1122
33
1122
33
5 (001) tilt boundary
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Steady State Grain Boundary Steady State Grain Boundary MigrationMigration
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Symmetric boundary
Asymmetric boundary = 14.04º
Asymmetric boundary = 26.57º
Bicrystal GeometryBicrystal Geometry
[010]
5 36.87º
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0 10 20 30 40 500
50
100
150
200
250
1400K 1200K 1000K
M (
10-9
m3 /N
s
• No mobility data available at a=0, 45º; zero driving force
• Mobilities vary by a factor of 4 over the range of inclinations studied at lowest temperature
• Variation increases when temperature ↓ (from ~2 to ~4)
• Minima in mobility occur where one of the boundary planes has low Miller indices
Mobility vs. InclinationMobility vs. Inclination
H. Zhang et al. Scripta Materialia, 52: 1193; 2005
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1.3
1.4
1.5
1.6
1.7
EG
B (
J/m
2 )
900K 1000K 1200K 1400K
10-14
10-13D
(cm
3 /s)
900K 1000K 1200K 1400K
• At low T, self-diffusivity & grain boundary energy increase with increasing inclination
• Mobility, self-diffusion coefficient and grain boundary energy exhibit local minimum at special inclination (at least one low index boundary plane)
• All three quantities are correlated for
a >18º
0 10 20 30 40 500
50
100
150
200
250
1400K 1200K 1000K
M (
10-9
m3 /N
s
(101)(001) (103)
N
2 2
i ii 1
GB
x yD
A 4t
N
GB i cohi 1
E E NE / A
Mobility, Diffusivity & Mobility, Diffusivity & EnergyEnergy
M. Mendelev et al. JMR, 20: 1146; 2005
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Cahn & Taylor’s Model Cahn & Taylor’s Model (2004)(2004)
• Boundary migration can also produce a coupled tangential motion of the two crystals relative to each other
• In the absence of grain boundary sliding, the velocity parallel to the grain boundary, v||, is proportional to the grain boundary migration velocity, vn. The coefficient is independent of grain boundary inclination.
• Coupling coefficient :
| || |
1 or =n
n
v uv s v
v M R
initial
pureshear
puresliding
combination
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Suzuki & Mishin’s Simulation Suzuki & Mishin’s Simulation (2005)(2005) v||
• [001] Symmetric tilt boundaries
• Fix the bottom and shear the top with v|| = 1m/s
• Grain boundary migrates ↑ or ↓
• 4
θ π θβ=2tan or -2tan -
2 2
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Shear Shear (coupled)(coupled) Motion - Symmetric Motion - Symmetric BoundaryBoundary
• 5 [010] symmetric tilt boundary (103) at 800K
• The step height = 1.11Ǻ ((103) plane spacing is 1.13Ǻ), therefore, the migration is plane by plane
• Both Ashby and Cahn give the correct prediction for symmetric grain boundary
v||=1m/s
CahnAshby
11.0
UR M
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Critical Stress for Shear Critical Stress for Shear (coupled)(coupled) MotionMotion
• When the shear strain of lower grain reaches ~0.4%, migration was ignited.
• The average critical stress is ~0.64 GPa.• This migration is difussionless
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Atomistic Migration Detail
• 12: Atomic configurations apart by ~122 ps
• The displacements represent elastic deformation; no indication of grain boundary sliding.
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Atomistic Migration Detail (Cont’d)
• 23: Atomic configurations apart by 5.6 ps
• Coupled sliding and migration shear
• Grain boundary migrates from blue line to red line
• Top crystal uniformly slides right – releases elastic strain
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Atomistic Jump Picture (23)
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v|| v|| v||
1 2 3
Macroscopic Migration Picture (Symmetric)
12: Elastic deformation, Stress ↑
23: Reach critical stress, two
grains slide relatively to each other;
stress release; boundary migrates
Fixed ratio of migration/sliding
shear
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Shear Motion in Asymmetric Boundaries
T=
50
0K
, v
||
=0
.5m
/s=9.46º =18.43
º
=26.57º
=36.87º
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T (K)
Cri
tical
Str
ess
(GP
a)
400 600 800 1000 12000.3
0.4
0.5
0.6
0.7
0.8
0.9
t (ps)
Bou
ndar
yP
ositi
on(Å
)
500 1000 1500 2000
74
75
76
77
78
79
80
81
82
T=1000KT=800KT=500K
Coupled motion at different T ( = 13.6º)
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Shear (Å)
Bou
ndar
yP
ositi
on(Å
)
0 5 10 15
60
70
80
90
100
= -9.9 = 0 = 8.6 = 18.4 = 31.9 = 37.2 = 45
Shear/coupled motion in General GBShear/coupled motion in General GB
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Critical StressesCritical Stresses