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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
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