MOLECULAR DYNAMICS SIMULATION OF STRESS INDUCED MOLECULAR DYNAMICS SIMULATION OF STRESS INDUCED GRAIN BOUNDARY MIGRATION IN NICKELGRAIN BOUNDARY MIGRATION IN NICKEL

MOLECULAR DYNAMICS SIMULATION OF STRESS INDUCED MOLECULAR DYNAMICS SIMULATION OF STRESS INDUCED GRAIN BOUNDARY MIGRATION IN NICKELGRAIN BOUNDARY MIGRATION IN NICKEL

Hao Zhang, Mikhail I. Mendelev, David J. SrolovitzDepartment of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08540

Background• Goal: Determine grain boundary mobility from

atomistic simulations

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

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

• boundary stiffness ” not readily available from atomistic simulations

• average over all inclinations

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

Molecular Dynamics

• Velocity Verlet

• Voter-Chen EAM potential for Ni

• Periodic BC in X, Y, free BC in Z

• Hoover-Holian thermostat and velocity rescaling

• 12,000 - 48,000 atoms, 0.5-10 ns

Application of Driving Force• Ideally, we want

• constant driving force during simulation• avoid NEMD • no boundary sliding

• Use elastic driving force• even cubic crystals are elastically anisotropic – equal

strain different strain energy• driving force for boundary migration: difference in

strain energy density between two grains

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

X

Y

Z

Grain Boundary

Free Surface

Free Surface

Grain

2G

rain 1

1122

33

1122

33

Linear Elastic Estimate of Driving Force

Non-symmetric tilt boundary

[010] tilt axis

boundary plane (lower grain) is (001)

Present case: 5 (36.8º)

Strain energy density

determine using linear elasticity

20

441211121144121244112

1111

2441211

212111211

)]4()2)(()2(6[

)2()2()2)((

CosCCCCCCCCCCCC

SinCCCCCCCF

)( 12 Grainelastic

Grainelastic FFMFMMpV

klijijklelastic CF 2

1

Conclusion• Developed new method that allows for the accurate

determination of grain boundary mobility as a function of misorientation, inclination and temperature

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

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

• The relation between driving force and applied strain2 and the relation between velocity and driving force are all non-linear

• Why is the velocity larger in tension than in compression?

ε

σ

*

• Strain energy density• Apply strain εxx=εyy=ε0 and σzz=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

Grain

1

Grain

2

0

0

1122 )(

dF Grainyy

Grainxx

Grainyy

Grainxx

• Typical strains• as large as 4% (Schönfelder et al.)• 1-2% here

Expand stress in powers of strain: ...2

11 BA

Non-Linear Stress-Strain Response

0 50000 100000 150000 200000 250000

50

55

60

65

70

75

1400K 1200K 800K

Gra

in B

ound

ary

Pos

itio

n (A

ngst

rom

)

Time Steps (10-14s)

• Fluctuations get larger as T ↑

GB Motion at Zero Strain

0 20000 40000 60000 80000 100000 120000 140000 16000040

45

50

55

60

Gra

in b

ound

ary

posi

tion

(A

ngst

rom

)

time steps (10-14s)

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

displacement)

Steady State Migration (Typical) Velocity vs. Driving Force

• Velocity under tension is larger than under compression (even after we account for elastic non-linearity)

• Difference decreases as T ↑

0.00 0.01 0.02 0.03 0.04-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Tensile Strain Compressive Strain

v (m

/s)

P (GPa)

0.00 0.01 0.02 0.03 0.04 0.05

0

1

2

3

4

5

6

Tensile Strain Compressive Strain

v (m

/s)

P (GPa)

0.00 0.01 0.02 0.03 0.04

0

1

2

3

4

5

6

Tensile Strain Compressive Strain

v (m

/s)

P (GPa)

0.00 0.01 0.02 0.03 0.04-1

0

1

2

3

4

5

6

7

8

Tensile Strain Compressive Strain

v (m

/s)

P (GPa)

800K

1200K 1400K

1000K

Mobility

• Activation energy for GB migration

is ~ 0.26 ±0.08eV

0.0007 0.0008 0.0009 0.0010 0.0011 0.0012 0.0013

1.52E-8

4.14E-8

1.13E-7

ln M

1/T (K-1)

Tp p

vM

lim0

p

v/p

Determination of Mobility

0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045 0.0500

10

20

30

40

50

60

70

80

90

100

110

120

Tensile Strain Compressive Strain

v/p

p

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

• Non-linear dependence of driving force on strain2

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

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

Driving Force

0.0000 0.0001 0.0002 0.0003 0.0004 0.0005

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09 800K T 800K C 1000K T 1000K C 1200K T 1200K C 1400K T 1400K C

P (

GP

a)

2

Non-Linear Driving Force

Implies driving force of form:

...3

1

2

1 3021

20210 BBAAP

-0.03 -0.02 -0.01 0.00 0.01 0.02 0.03

-15

-10

-5

0

5

10

Upper Grain Bottom Grain

xx

+yy

(GPa)

0.0000 0.0001 0.0002 0.0003 0.0004 0.0005

0.00

0.01

0.02

0.03

0.04

0.05

0.00040 0.00045 0.00050

0.04

0.05

800K Tension 800K Conpression Linear Elasticity

2

P (

GP

a)