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