princeton university department of mechanical and aerospace engineering stress-driven grain boundary...
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Princeton UniversityDepartment of Mechanical and Aerospace Engineering
Stress-Driven Grain Boundary Migration
Effect of Boundary Inclination on Mobility
Hao Zhang; Mikhail I. Mendelev; David J. Srolovitz
Motivation
• Quantitative intrinsic grain boundary mobility data are difficult
to obtain from experiment, but important for predicting
microstructural evolution
• Capillarity driven boundary motion is useful, but has limitations
• yields reduced mobility M*=M(+ ”) instead of M
• boundary stiffness (+ ”) was never measured
• simulations give reduced mobility averaged over all inclinations
• Elastic stresses can be used to drive the motion of flat grain
boundaries
• Easy to measure GB mobility of fully-crystallographically
defined boundaries
Stress Driven Grain Boundary Motion
• Ideally, we want• constant driving force during simulation• avoid NEMD (Schönfelder et al.)• no boundary sliding
• 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
X
Y
Z
Grain Boundary
Free Surface
Free Surface
Grain
2G
rain 1
1122
33
1122
33
5 (001) tilt boundary
Steady State Grain Boundary Migration
Non-Linear Stress-Strain Response
ε
σ
...211 BA
ε*
Grain
1
Grain
2• Typical strains
• as large as 4% (Schönfelder et al.)• 1-2% here
• 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:
0
0
1122 )(
dF Grainyy
Grainxx
Grainyy
Grainxx
• Driving force
Non-Linear Driving Force
Implies driving force of form:
...3
1
2
1 3021
20210 BBAAP
-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)
0 1 2 3 4 5
0.00
0.01
0.02
0.03
0.04
0.05
0.00040 0.00045 0.00050
0.04
0.05
800K Tension 800K Conpression Linear Elasticity
2 (10-4)
P (
GP
a)• Non-linear dependence of driving force on strain2
• Driving forces are larger in tension than compression for same strain
• Compression and tension give same driving force at small strain (linearity)
Determination of Mobility
Tp p
vM
lim0
p
v/p
0.00 0.01 0.02 0.03 0.04 0.050
40
80
120
Tensile Strain Compressive Strain
v/p
p
• Determine mobility by extrapolation to zero driving force
• Tension (compression) data approaches from above (below)
• Activation energy for GB migration is ~ 0.26 ±0.08 eV
• Simulations using a half-loop geometry (same misorientation) give the same activation energy
Activation Energy for GB Migration
0.6 0.8 1.0 1.2 1.4
2.7E-8
7.4E-8
2E-7ln
M
1/T ( x1000 K-1)
[010]
5 36.87ºSymmetric boundary
Asymmetric boundary = 14.04º
Asymmetric boundary = 26.57º
Simulation / Bicrystal Geometry
All simulations performed at fixed misorientation at 1200K
Mobility Dependence on Boundary Inclination
• Mobilities vary by a factor of 3.5 over the range of inclinations studied
• Minima in mobility occur when one of the boundary planes has low Miller indices
Inclination º
Bottom Grain
Normal Plane
Top Grain Normal Plane
0 (1 0 3) (1 0 )
9.46 (9 0 17) ( 0 19)
11.31 (4 0 7) ( 0 8)
14.04 (7 0 11) ( 0 13)
18.43 (3 0 4) (0 0 1)
21.80 (11 0 13) (1 0 17)
26.57 (1 0 1) (1 0 7)
30.96 (7 0 6) (2 0 9)
36.87 (13 0 9) (1 0 3)
3
3
1
1
(001)
(103)
(101)
0 10 20 30 40
40
80
120
160
200
M (1
0-9 m
3 /Ns)
()
GB Diffusivity Dependence on Inclination
• There is no correlation between grain boundary diffusivity and mobility
0 10 20 30 40
1.0
1.2
1.4
1.6
D (1
0-13 m
3 /s)
() 0 10 20 30 40
40
80
120
160
200
M (1
0-9 m
3 /Ns)
()
Capillarity Driven Grain Boundary Motion
gbv M F
''gb gbF
''gb gb gbv M
• capillarity-driven migration
''gb gb gbM M
32 / 4gV N a*Mz z
2gdVvwz M wz M z
dt
• FCC Nickel <001> 5Tilt Grain Boundary
• Voter-Chen EAM – Ni
w
• Extract reduced mobility from the rate of change of half-loop volume
0.0007 0.0008 0.0009 0.0010
4.1E-8
ln M
*
1/T (K-1)
Simulation Results
• Activation energy is 0.26 ±0.02eV
• This is the same activation energy found in flat boundary migration for this misorientation
• Steady-state migration behavior
• Slope proportional to reduced mobility
Conclusion
• Developed new method (stress driven GB motion) to determine grain boundary mobility as a function of , and T
• Non-linearities in elasticity and velocity-driving force relation are significant at large strain
• Activation energy is small, 0.26 eV in Ni
• Grain boundary mobility varies by a factor of 3.5 with inclination at 0.75Tm
• Minima in boundary mobility occurs where at least one boundary plane is a low index plane
• No correlation between grain boundary diffusivity and mobility
• Activation energies for grain boundary migration obtained by stress and capillarity driven are similar