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```					  Stress Driven Migration of
Flat Grain Boundaries

Hao Zhang, Mikhail I. Mendelev and David J. Srolovitz
Princeton University
Outline
• Motivation
• Elastic Driving Forces
• Simulation Method
• Simulation Results
• Driving Force vs. Strain
• Grain Boundary velocity vs. Driving Force at different
temperature
• Activation Energy

• Conclusion
Motivation
• Want to extract grain boundary mobility from
atomistic simulations
• Half-Loop models are useful, but not
sufficient
• yields the reduced mobility M*=M(g+ g”) not M
itself

• boundary stiffness g+g” is difficult to accurately
determine from atomistic simulations                                        

M (g + g   M * 
Ag
• reduced mobility is a average value over all                                
inclinations

• Flat boundary geometry can be used to
directly determine mobility (Schönfelder, et al.)
Simulation Method
• Molecular Dynamics
• Velocity Verlet

• Voter-Chen EAM potential for Ni

• Periodic BC in X, Z, free in Y-directions

• 12,000 - 48,000 atoms, 0.5-10 nanoseconds

• Hoover-Holian thermostat and velocity rescaling
How Do We Apply a Driving Force?
• Want
•
Y
constant driving force during simulation
•   avoid NEMD                                               X

•   no boundary sliding                             Z         Free
Surface
•   single boundary                                                              q

Grain 2
• Use elastic driving force
• even cubic crystals are elastically anisotropic            Grain

– equal strain  different strain energy                  Boundary

• driving force for boundary migration:

Grain 1
22
difference in strain energy density between                    33
11

two grains                                                 Free
Surface

• Apply strain
• apply biaxial strain in x and z, free surface
normal to y
Driving Force
Need accurate determination of driving force
Non-symmetric tilt boundary                               Y
Free
Surface
[001] tilt axis                                             X                               q

Grain 2
boundary plane (lower grain) is (010)            Z

Present case: S5 (36.8º)                                                    Grain
Boundary

Strain energy density

Grain 1
22
11
determine using linear elasticity                                          33

1                                              Free
Felastic  Cijkl  ij  kl                              Surface
2
V  Mp  MF  M ( Felastic 2  Felastic 1 )
Grain        Grain

(C11  C12 )(C11 + 2C12 ) 2 (C11  C12  2C44 )[Cos(4q )  1]
F                                                                                      02
2C11[C11  6C11C44 + C12 (C12 + 2C44 ) + (C11 + C12 )(C11  C12  2C44 )Cos(4q )]
2
Non-Linear Stress-Strain Response
• Typical strains
σ                       • as large as 4% (Schönfelder et al.)
• 1-2% here
• Strain energy density
• Apply strain εxx=εzz=ε0 and σyy=0
to perfect crystals, measure stress
ε*    ε            vs. strain and integrate to get the
strain contribution to free energy
• Includes non-linear contributions to
elastic energy
0
F (    ( xx 2 +  zz 2   xx 1   zz 1 )d 
Grain    Grain    Grain    Grain
0

Grain1

Grain2
Expand stress in powers of strain:
 (   A1 + B1 2 + ...
Non-Linear Driving Force
15
(GPa)
Grain2
10
Grain1

5

0.06

0
Tension
-0.03     -0.02      -0.01         0.00      0.01   0.02   0.03 0.05
                                        Compression
-5

Drving Force (GPa)
0.04

-10
0.03

-15                                                    0.02

0.01

Implies driving force of form:                                              0.00
0.0000   0.0001    0.0002           0.0003   0.0004   0.0005

P( 0            ( A1  A2  0 2 + 1 (B1  B2  03 + ...
1                                                                                           Strain
2

2                    3
Driving Force

• Non-linear
800K T
0.09
800K C                                                 dependence of
1000K T                                                driving force on
0.08
1000K C                                                strain
0.07        1200K T
1200K C                                              • Driving forces are
Driving Force (Gpa)

0.06        1400K T                                                larger in tension
14000 C
0.05
than compression
for same strain (up
0.04                                                               to 17% at 0=0.02)
0.03
• Compression and
0.02                                                               tension give same
driving force at
0.01
small strain
0.00                                                               (linearity)
0.0000    0.0001   0.0002          0.0003   0.0004   0.0005
2
Strain
Grain Boundary Motion at Zero Strain

75
Grain Boundary Position (Angstrom)

1400K
1200K
70
800K

65

60

55

50

0        50000       100000     150000   200000   250000
-14
Time Steps (10 s)

• Fluctuations get larger as T ↑
Steady State Migration – Low Driving Force
60

Grain boundary position (Angstrom)

55

50

45

40
0   20000 40000 60000 80000 100000 120000 140000 160000
-14
time steps (10 s)
• At high T, fluctuations can be large
• Determine mobility based upon large boundary displacement
Velocity vs. Driving Force
800K                                                                     1000K
6
4.0

3.5                                                                        5
Tensile Strain                                                          Tensile Strain
3.0           Compressive Strain                                                      Compressive Strain
4
2.5

Velocity (m/s)
Velocity (m/s)

2.0                                                                        3

1.5
2
1.0

1
0.5

0.0
0
-0.5
0.00    0.01      0.02      0.03     0.04   0.05                        0.00       0.01       0.02       0.03   0.04   0.05
Driving Force (GPa)                                                        Driving Force (GPa)
Velocity vs. Driving Force (Continued)
1200K                                                                            1400K
6
8

5                                                                           7
Tensile Strain                                                              Tensile Strain
Compressive Strain                                              6           Compressive Strain
4
5

Velocity (m/s)
Velocity (m/s)

3                                                                           4

3
2
2

1                                                                           1

0
0
-1
0.00       0.01       0.02       0.03   0.04   0.05                         0.00        0.01           0.02          0.03   0.04
Driving Force (GPa)                                                          Driving Force (GPa)

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

 v               120
M  lim                   110
p 0  p T             100
Tensile Strain
Compressive Strain
90

80
v/p

70

60

v/p
50

40

30

20

10

0

p          0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045 0.050
p
Activation Energy for GB Migration

• Activation energy
-15.6

-15.8                                                                      for GB migration is
-16.0
~ 0.2 ±0.016eV

-16.2
ln M

-16.4

-16.6

-16.8

-17.0
0.0007   0.0008   0.0009    0.0010    0.0011   0.0012   0.0013
-1
1/T (K )
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 0.2 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 velocity larger at large strain larger in tension than
compression?

```
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 views: 0 posted: 3/2/2013 language: English pages: 17