MOLECULAR DYNAMICS SIMULATION OF STRESS INDUCED GRAIN BOUNDARY

MOLECULAR DYNAMICS SIMULATION OF STRESS INDUCED GRAIN BOUNDARY

MOLECULAR DYNAMICS SIMULATION OF STRESS INDUCED
GRAIN BOUNDARY MIGRATION IN NICKEL
Hao Zhang, Mikhail I. Mendelev, David J. Srolovitz
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08540
Molecular Dynamics

Goal: Determine grain boundary mobility from
atomistic simulations

Application of Driving Force
Ideally, we want

Velocity Verlet
Periodic BC in X, Y, free BC in Z

0

F (
0

Grain1
xx

0.00

0.01

0.02

0.03

0.00040

800K Tension
800K Conpression
Linear Elasticity

0.0000

0.0001

1
1
2
3
P 0 A1 A2 0 B1 B2 0 ...
2
3

v (m/s)

1.5

0.0002

0.0003

0.0000

v
M lim
p 0 p T

3

2

0.01

100

0.04

0.05

Tensile Strain
Compressive Strain

1400K

8

5

v (m/s)

2

1

0

20000 40000 60000 80000 100000 120000 140000 160000
-14

time steps (10 s)

At high T, fluctuations can be large
Velocity from mean slope
Average over long time (large boundary
displacement)

-1
0.00

0.01

0.02

P (GPa)

0.03

0.04

4.14E-8

0.00

0.01

0.02

0.03

0.04

p

p

0.0007

0.0008

0.0009

0.0010

0.0011

0.0012

0.0013

-1

1/T (K )

P (GPa)

Velocity under tension is larger than under compression
(even after we account for elastic non-linearity)
Difference decreases as T

70

65

60

55

50
0

50000

100000

150000

200000

250000

-14

Time Steps (10 s)

Fluctuations get larger as T

Activation energy is much smaller than found in
experiment (present results 0.26 eV in Ni, experiment
2-3 eV in Al)

1.52E-8

0
0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045 0.050

0

0

60

10

1

1400K
1200K
800K

Activation energy for grain boundary migration is
finite; grain boundary motion is a thermally activated
process

1.13E-7

20

2

40

Tensile Strain
Compressive Strain

30

3

75

Developed new method that allows for the accurate
determination of grain boundary mobility as a
function of misorientation, inclination and
temperature

40

4

GB Motion at Zero Strain

Conclusion

Mobility

50

Tensile Strain
Compressive Strain

6

3

0.0005

70

7

4

v (m/s)

0.03

v/p

1200K

5

45

0.02

P (GPa)

P (GPa)
6

0.0004

80
0.00

v/p

50

0.0003

110

-0.5
0.04

0.0002

120

0
0.03

0.0001

2

90

0.02

0.00

2

1

0.0

0.0005

Tensile Strain
Compressive Strain

1.0
0.5

0.0004

Compression and
tension give same
driving force at
small strain
(linearity)

1000K

5

4

2.0

0.02

Determination of Mobility

6

Tensile Strain
Compressive Strain

0.01

0.05

0.01

Implies driving force of form:

800K

0.00

0.06

Driving forces are
larger in tension
than compression
for same strain (up
to 13% at 0=0.02)

0.03

0.00

2.5

v (m/s)

0.00050

0.01

)d

3.5

55

0.00045

0.02

-10

Velocity vs. Driving Force
3.0

Non-linear
dependence of
driving force on
strain2

0.04

4.0

Grain boundary position (Angstrom)

0.03

-15

60

0.07

P (GPa)

-0.01
-5

Grain1
yy

0.08

0.04

0
-0.02

(C11 C12 )(C11 2C12 ) 2 (C11 C12 2C44 ) Sin2 (2 )
2
F
0
2
C11[C11 6C11C44 C12 (C12 2C44 ) (C11 C12 )(C11 C12 2C44 )Cos(4 )]

800K T
800K C
1000K T
1000K C
1200K T
1200K C
1400K T
1400K C

0.09

0.04

Grain2

Steady State Migration (Typical)

Grain 2
yy

Grain1

Expand stress in powers of
strain:
A1 B1 2 ...

Grain 2
xx

0.05

1
Felastic Cijkl ij kl
2

Grain 2
Grain1
V Mp MF M ( Felastic
Felastic
)

0.05

5

-0.03

11

Free
Surface

ln M

*

xx+yy (GPa)

P (GPa)

22

determine using linear elasticity

Driving Force

Upper Grain
Bottom Grain

Apply strain xx=yy=0 and zz=0
to perfect crystals, measure
stress vs. strain and integrate to
get the strain contribution to
free energy
Includes non-linear
contributions to elastic energy

Strain energy density

33

Non-Linear Driving Force
10

Present case: 5 (36.8))

Grain
Boundary

apply constant biaxial strain in x and y
free surface normal to z provides zero stress in z

Strain energy density

boundary plane (lower grain) is (001)

Grain 1

as large as 4% (Schnfelder et al.)
1-2% here

33

Apply strain

Flat boundary geometry can be used to directly
determine mobility, but subtle (Schnfelder, et al.)

11

22

even cubic crystals are elastically anisotropic equal
strain different strain energy
driving force for boundary migration: difference in
strain energy density between two grains

12,000 - 48,000 atoms, 0.5-10 ns

Non-Linear Stress-Strain Response

[010] tilt axis

Free
Surface

Y

Use elastic driving force

average over all inclinations

Typical strains

X

Grain 2

boundary stiffness not readily available from
atomistic simulations

Non-symmetric tilt boundary

Hoover-Holian thermostat and
velocity rescaling

gives reduced mobility, M =M ), rather than M
*

Z

constant driving force during simulation
avoid NEMD
no boundary sliding

Voter-Chen EAM potential for Ni

Methods based upon capillarity driving force
are useful, but not sufficient

Linear Elastic Estimate of Driving Force

Grain Boundary Position (Angstrom)

Background

Determine mobility by extrapolation to zero driving force
Tension (compression) data approaches from above (below)

Activation energy for GB migration
is ~ 0.26 0.08eV

The relation between driving force and applied strain 2
and the relation between velocity and driving force
are all non-linear
Why is the velocity larger in tension than in
compression?

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