Determination of

Grain Boundary Stiffness

Hao Zhang1, Mikhail Mendelev1,2 and David Srolovitz1

1PRISM, Princeton University

2Ames Laboratory

Driving Force

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, xx = yy = 0

• 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

Real 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 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:• Driving force

0

0

1122 )(

dF Grainyy

Grainxx

Grainyy

Grainxx

• Implies driving force of form:

2 30 1 2 0 1 2 0

1 1...

2 3F A A B B

-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)

Zhang H, Mendelev MI, Srolovitz DJ. Acta Mater 52:2569 (2004)

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. Inclinations

• 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 indices0 10 20 30 40 50

0

50

100

150

200

250

1400K 1200K 1000K

Mob

ility

(1

0-9 m

3 /Ns)

(101)(001) (103)

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 migration with inclination is significant

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

• Activation energy for symmetric boundaries, ? ? ?

0 10 20 30 40 50

0.1

0.2

0.3

0.4

0.5

Q (

eV)

O

rn

*,, , ,r n

R tv v M R t R t

t

r n r n n

* ''M M n

Determination of Grain Boundary Stiffness

• Capillarity driven migration

*"v M M

• Determine reduced mobility from simulation of shrinking, grain

2 2

3/ 22 2

2,

R R RRt

R R

2 2

R RR r rR R R

r r

r θn

• Radial velocity for arbitrary curve<010>

<100>

<010>

<100>

*0

0

,

1 14 cos 4

R t M

t R t

f

n

• Keep the first order terms

*

2 2 30

0

,1 1 14 cos 4 257cos 4 16

R tMf O

R t t

n

• Substituting into expression for the velocity and rearranging terms

24 sin 4 O

0, 1 cos 4R t R t

* *0, , 1M n R t M f n

• If grain shape is only slightly different from a circle, we can assume

0

2

0

4cos sin 4 sin

cos =4

1 sin 4

R t

R

R t

R

• To find how the reduced mobility varies with inclination, , we must relate to

Determination of Grain Boundary Stiffness (Contd)

4-fold symmetry - [010] tilt

O

rn

Circular Shrinkage Geometry

0.0 0.1 0.2 0.3 0.4 0.50

2

4

6

8

10

12

0.2 0.3 0.4 0.5 0.6

0

2

4

6

8

10

12

1400K

R02 (1

0-17 m

2 )

t (ns)

500K

Simulation Result

• Steady-state migration during circular shrinkage

• Migration velocity strongly depends on temperature

• Activation energy for migration is 0.2eV

0.7 0.8 0.9 1.0

5E-8

1E-7

ln M

0*1/T (0.001K-1)

0

30

6090

120

150

180

210

240270

300

330

0.98

1.00

0.98

1.00

Simulation Result 1+cos4

±0.003

0

0

, where is an arbitrary constantR R

C CR

Circular Shape

is temperature independent between 1000 and 1400 K to within the accuracy of these simulations

assumed functional form of grain shape is in agreement with simulation results

Stiffness vs. Inclination

• At high temperature, Stiffness is not significantly changed with inclinations

• General speaking, stiffness is larger at low T than at high T

• The ratio of maximum to minimum at 1000K is ~3

• Can not determine the existence of cups around the two symmetric grain boundaries

10 20 30 400.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

'' (J

/m2 )

()

1000K 1200K 1400K

Using M from the flat boundary simulations and M* from the shrinking grain simulations, we determine stiffness vs. boundary inclination

Conclusion

• Developed new method (stress driven GB motion) to determine

grain boundary mobility as a function of , and T

• Extracted grain boundary stiffness from atomistic simulations

• Mobility is a strong function of inclination and temperature;

mobility exhibits minima where at least one of the boundary

planes has low Miller indices

• Grain boundary stiffness varies with inclination and is only

weakly temperature-dependent