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

• 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

• Reduced mobility increases with increasing temperature

• Mobility shows maxima at low Σ misorientations

Reduced Mobility vs. MisorientationReduced Mobility vs. Misorientation

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

Steady State Grain Boundary Steady State Grain Boundary MigrationMigration

Symmetric boundary

Asymmetric boundary = 14.04º

Asymmetric boundary = 26.57º

Bicrystal GeometryBicrystal Geometry

[010]

5 36.87º

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

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

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

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

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

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

Atomistic Migration Detail

• 12: Atomic configurations apart by ~122 ps

• The displacements represent elastic deformation; no indication of grain boundary sliding.

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

Atomistic Jump Picture (23)

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

Shear Motion in Asymmetric Boundaries

T=

50

0K

, v

||

=0

.5m

/s=9.46º =18.43

º

=26.57º

=36.87º

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)

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

Critical StressesCritical Stresses