Dependence of Grain Boundary Mobility on Boundary Plane

Hao Zhang1, Mikhail Mendelev1,2 and David Srolovitz1

1PRISM, Princeton University

2Ames Laboratory

Challenges

• Neither curvature driven boundary migration experiments nor simulations yield the fundamental kinetic properties for grain boundary migration

• , M* is the product of the mobility and grain boundary stiffness

• Reduced mobility is averaged over all possible inclinations

• The migration of a flat boundary is easier to analyze, but has several limitations

• Can yield grain boundary mobility dependence on inclination

• Is the variation of grain boundary mobility correlated with other boundary properties, such as grain boundary energy and self-diffusivity?

*"v Mp M M

Elastically-Driven Migration of a Flat Boundary

X

Y

Z

Grain Boundary

Free Surface

Free Surface

Grain

2G

rain 1

1122

33

1122

33

5 (001) tilt boundary

• 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

• Driving Force based on linear Elasticity

20

441211121144121244112

1111

4412112

12111211

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

]1)4()[2()2)((

CosCCCCCCCCCCCC

CosCCCCCCCF

2 1( )Grain Grainelastic elasticv Mp M F M F F

klijijklelastic CF 2

1

Measured Driving Force

...211 BA

Grain

1

Grain

2

• Typical strains•1-2%, out of linear region

• Measuring driving force• Apply strain εxx=εyy=ε0 and σiz= 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:• Driving force

0

0

1122 )(

dF Grainyy

Grainxx

Grainyy

Grainxx

-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)

• Implies driving force of form:

2 30 1 2 0 1 2 0

1 1...

2 3F A A B B

Determination of Mobility

Tp p

vM

lim0

p

v/p

• Determine mobility by extrapolation to zero driving force

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

0.00 0.01 0.02 0.03 0.0440

80

120

160

200

v/p

p

Tensile Strain Compressive Strain

Symmetric boundary

Asymmetric boundary = 14.04º

Asymmetric boundary = 26.57º

Simulation / Bicrystal Geometry

[010]

5 36.87º

Initial Simulation Cell for Different Inclinations

Mobility vs. Inclination

0 10 20 30 40 500

50

100

150

200

250

1400K 1200K 1000K

Mob

ility

(1

0-9 m

3 /Ns)

• No mobility data available at =0, 45º; zero biaxial strain driving force

• Mobilities vary by a factor of 4 over the range of inclinations studied at lowest temperature

• Variation decreases when temperature ↑ (from ~4 to ~2)

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

Activation Energy vs. Inclination

Tk

QMM

B

exp0

0.1 0.2 0.3 0.4 0.5

-14

-13

-12

-11

Q (eV)ln

M0(m

3 /Ns)

• The variation of activation energy for grain boundary migration over the inclination region we studied is significant

• The variation of mobility becomes weaker than expected on the basis of activation energy because of the compensation effect

• Activation energy for the symmetric boundary is unknown

0 10 20 30 40 50

0.1

0.2

0.3

0.4

0.5

Q (

eV)

Diffusivity vs. Inclination

2 2

1

4

GBN

i ii

GB

x yD

A t

0.7 0.8 0.9 1.0 1.1

-46

-44

-42

-40

-38

-36

-34

-32

-30

-28

ln D

(cm

3 /s)

1/T ( 1/K)

18 14 11 9 0 22 26 31 36 45

0 10 20 30 40 50

10-14

10-13

D (

cm3 /s

)

()

900K 1000K 1200K 1400K

• Diffusivity shows more anisotropic at low temperature than at high temperature

• Most of local minimum corresponds to one of the grains normal with low Miller indices

• The =0º has a change from minimum to maximum

Activation Energy and Compensation Effect

0 10 20 30 40 500.4

0.6

0.8

1.0

1.2

Q (

eV)

()0.4 0.6 0.8 1.0 1.2

-25

-24

-23

-22

-21

-20

-19

ln D

0Q (eV)

• The activation energy all lie between 0.5 to 0.6 eV, except for the º symmetric boundary(1.1 eV)

• Compensation effect weaken the diffusivity variation based upon the activation energy for self-diffusion

Mobility, Self-diffusion and Energy

1.3

1.4

1.5

1.6

1.7

GB

Ene

rgy

(J/m

2 )

900K 1000K 1200K 1400K

10-14

10-13D

(cm

3 /s)

900K 1000K 1200K 1400K

• At low temperature, self-diffusion and grain boundary energy have similar trend, i.e. change from minimum to maximum, but mobility has opposite trend.

• Mobility, self-diffusion coefficient and grain boundary energy shows local minimum at special inclination (one of the plane normal is low Miller indices)

• There exists correlation between those three quantities in the inclination range of 18º to 45º.

0 10 20 30 40 500

50

100

150

200

250

1400K 1200K 1000K

Mob

ility

(1

0-9 m

3 /Ns)

(101)(001) (103)

tA

yxD

N

iii

GB 41

22

ANEEE

N

icohiGB /

1

Conclusion

• Used stress driven GB motion to determine grain boundary mobility

as a function of , and T

• Mobility is a strong function of inclination and temperature

• Grain boundary self-diffusion is sensitive to inclinations, i.e. grain

boundary structure

• Minima in boundary mobility, self-diffusion coefficient and grain

boundary energy occurs where at least one boundary plane is a low

index plane

• In the inclination range from 18º to 45º, there is a strong correlation

between grain boundary diffusivity, energy and mobility