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?