Hao Zhang 1 , David J. Srolovitz 1,2 1 Princeton University 2 Yeshiva University
Dependence of Grain Boundary Mobility on Boundary Plane Hao Zhang 1, Mikhail Mendelev 1,2 and David...
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Transcript of Dependence of Grain Boundary Mobility on Boundary Plane Hao Zhang 1, Mikhail Mendelev 1,2 and David...
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