Princeton UniversityDepartment of Mechanical and Aerospace Engineering

The Effect of Inclination

on Grain Boundary Mobility

Hao Zhang; Mikhail I. Mendelev; David J. Srolovitz

Motivation

• Quantitative intrinsic grain boundary mobility data are difficult

to obtain from experiment, but important for predicting

microstructural evolution

• Capillarity driven boundary motion is useful, but has limitations

• yields reduced mobility M*=M(+ ”) instead of M

• boundary stiffness (+ ”) was never measured

• simulations give reduced mobility averaged over all inclinations

• Elastic stresses can be used to drive the motion of flat grain

boundaries

• Easy to measure GB mobility of fully-crystallographically

defined boundaries

Stress Driven Grain Boundary Motion

• Ideally, we want• constant driving force during simulation• avoid NEMD (Schönfelder et al.)• 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

• Applied strain• constant biaxial strain in x and y• free surface normal to z iz = 0

X

Y

Z

Grain Boundary

Free Surface

Free Surface

Grain

2G

rain 1

1122

33

1122

33

Steady State Grain Boundary Migration

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

11 12 44( 2 )aC C C C

)( 12 Grainelastic

Grainelastic FFMFMMpV

klijijklelastic CF 2

1

X

Y

Z

Grain Boundary

Free Surface

Free Surface

Grain

2G

rain 1

1122

33

1122

33

2 2

211 12 11 120

11 11 11 12 11 12

( )( 2 ) sin (2 )

4 ( ) ( ) 1 cos(4 )a

a a

C C C C CF

C C C C C C C C

Non-Linear Stress-Strain Response

ε

σ

...211 BA

ε*

Grain

1

Grain

2• Typical strains

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

• Measuring Driving force• 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

• Fit stress:

0

0

1122 )(

dF Grainyy

Grainxx

Grainyy

Grainxx

• Driving force

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 1 2 3 4 5

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 (10-4)

P (

GP

a)• Non-linear dependence of driving force on strain2

• Driving forces are larger in tension than compression for same strain

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

Velocity vs. Driving Force

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

• Difference decreases as T ↑

800K

1200K 1400K

1000K

0.00 0.01 0.02 0.03 0.040

1

2

3

4

Tensile Strain Compressive Strain

v (m

/s)

P (GPa)

0.00 0.01 0.02 0.03 0.040

1

2

3

4

5

6

Tensile Strain Compressive Strain

v (m

/s)

P (GPa)

0.00 0.01 0.02 0.03 0.040

1

2

3

4

5

6

Tensile Strain Compressive Strain

v (m

/s)

P (GPa)

0.00 0.01 0.02 0.03 0.040

1

2

3

4

5

6

7

8

Tensile Strain Compressive Strain

v (m

/s)

P (GPa)

Determination of Mobility

Tp p

vM

lim0

p

v/p

0.00 0.01 0.02 0.03 0.04 0.050

40

80

120

Tensile Strain Compressive Strain

v/p

p

• Determine mobility by extrapolation to zero driving force

• Tension (compression) data approaches from above (below)

• Activation energy for GB migration is ~ 0.26 ±0.08 eV

• Simulations using a half-loop geometry (same misorientation) give the same activation energy

Activation Energy for GB Migration

0.6 0.8 1.0 1.2 1.4

2.7E-8

7.4E-8

2E-7ln

M

1/T ( x1000 K-1)

[010]

5 36.87ºSymmetric boundary

Asymmetric boundary = 14.04º

Asymmetric boundary = 26.57º

Simulation / Bicrystal Geometry

All simulations performed at fixed misorientation at 1200K

Mobility Dependence on Boundary Inclination

• Mobilities vary by a factor of 3.5 over the range of inclinations studied

• Minima in mobility occur when one of the boundary planes has low Miller indices

Inclination º

Bottom Grain

Normal Plane

Top Grain Normal Plane

0 (1 0 3) (1 0 )

9.46 (9 0 17) ( 0 19)

11.31 (4 0 7) ( 0 8)

14.04 (7 0 11) ( 0 13)

18.43 (3 0 4) (0 0 1)

21.80 (11 0 13) (1 0 17)

26.57 (1 0 1) (1 0 7)

30.96 (7 0 6) (2 0 9)

36.87 (13 0 9) (1 0 3)

3

3

1

1

(001)

(103)

(101)

0 10 20 30 40

40

80

120

160

200

M (1

0-9 m

3 /Ns)

()

GB Diffusivity Dependence on Inclination

• So far, there is no strong correlation between grain boundary diffusivity and mobility

0 10 20 30 40

40

80

120

160

200

M (1

0-9 m

3 /Ns)

()0 10 20 30 40

1.0

1.2

1.4

1.6

1.0

2.0

3.0

4.0

5.0

6.0

7.0

1200K

(1

0-14 m

3 /s)

D (1

0-13 m

3 /s)

()

1000K

Capillarity Driven Grain Boundary Motion

gbv M F

''gb gbF

''gb gb gbv M

• capillarity-driven migration

''gb gb gbM M

32 / 4gV N a*Mz z

2gdVvwz M wz M z

dt

• FCC Nickel <001> 5Tilt Grain Boundary

• Voter-Chen EAM – Ni

w

• Extract reduced mobility from the rate of change of half-loop volume

0.0007 0.0008 0.0009 0.0010

4.1E-8

ln M

*

1/T (K-1)

Simulation Results

• Activation energy is 0.26 ±0.02eV

• This is the same activation energy found in flat boundary migration for this misorientation

• Steady-state migration behavior

• Slope proportional to reduced mobility

Conclusion

• Developed new method (stress driven GB motion) to determine grain boundary mobility as a function of , and T

• Non-linearities in elasticity and velocity-driving force relation are significant at large strain

• Activation energy is small, 0.26 eV in Ni

• Grain boundary mobility varies by a factor of 3.5 with inclination at 0.75Tm

• Minima in boundary mobility occurs where at least one boundary plane is a low index plane

• Correlation between grain boundary diffusivity and mobility?

• Activation energies for grain boundary migration obtained by stress and capillarity driven are similar

Outline

• Motivation

• Stress Driven Grain Boundary Motion

• Elastic Driving Force

• Simulation Method

• Simulation Results

• Temperature Effect

• Inclination Effect

• Comparison with Capillarity-Driven Motion & Experiment

• Correlation with Grain Boundary Diffusivity