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					  Stress Driven Migration of
     Flat Grain Boundaries

Hao Zhang, Mikhail I. Mendelev and David J. Srolovitz
                Princeton University
Outline
• Motivation
• Elastic Driving Forces
• Simulation Method
• Simulation Results
   • Driving Force vs. Strain
   • Steady State Migration
   • Grain Boundary velocity vs. Driving Force at different
     temperature
   • Activation Energy

• Conclusion
Motivation
• Want to extract grain boundary mobility from
  atomistic simulations
• Half-Loop models are useful, but not
  sufficient
    • yields the reduced mobility M*=M(g+ g”) not M
      itself

    • boundary stiffness g+g” is difficult to accurately
      determine from atomistic simulations                                        

                                                           M (g + g   M * 
                                                                                  Ag
    • reduced mobility is a average value over all                                
      inclinations

• Flat boundary geometry can be used to
  directly determine mobility (Schönfelder, et al.)
Simulation Method
• Molecular Dynamics
  • Velocity Verlet

  • Voter-Chen EAM potential for Ni

  • Periodic BC in X, Z, free in Y-directions

  • 12,000 - 48,000 atoms, 0.5-10 nanoseconds

  • Hoover-Holian thermostat and velocity rescaling
  How Do We Apply a Driving Force?
• Want
   •
                                                           Y
       constant driving force during simulation
   •   avoid NEMD                                               X


   •   no boundary sliding                             Z         Free
                                                                Surface
   •   single boundary                                                              q




                                                                               Grain 2
• Use elastic driving force
   • even cubic crystals are elastically anisotropic            Grain

     – equal strain  different strain energy                  Boundary


   • driving force for boundary migration:




                                                                               Grain 1
                                                                     22
     difference in strain energy density between                    33
                                                                          11



     two grains                                                 Free
                                                               Surface

• Apply strain
   • apply biaxial strain in x and z, free surface
       normal to y
Steady State Grain Boundary Migration
     Driving Force
      Need accurate determination of driving force
      Non-symmetric tilt boundary                               Y
                                                                                   Free
                                                                                  Surface
           [001] tilt axis                                             X                               q




                                                                                                  Grain 2
           boundary plane (lower grain) is (010)            Z



      Present case: S5 (36.8º)                                                    Grain
                                                                                 Boundary

      Strain energy density




                                                                                                  Grain 1
                                                                                        22
                                                                                             11
           determine using linear elasticity                                          33

                                    1                                              Free
                          Felastic  Cijkl  ij  kl                              Surface
                                    2
                   V  Mp  MF  M ( Felastic 2  Felastic 1 )
                                        Grain        Grain


                (C11  C12 )(C11 + 2C12 ) 2 (C11  C12  2C44 )[Cos(4q )  1]
F                                                                                      02
     2C11[C11  6C11C44 + C12 (C12 + 2C44 ) + (C11 + C12 )(C11  C12  2C44 )Cos(4q )]
              2
   Non-Linear Stress-Strain Response
                                     • Typical strains
                  σ                       • as large as 4% (Schönfelder et al.)
                                          • 1-2% here
                                     • Strain energy density
                                         • Apply strain εxx=εzz=ε0 and σyy=0
                                           to perfect crystals, measure stress
                        ε*    ε            vs. strain and integrate to get the
                                           strain contribution to free energy
                                         • Includes non-linear contributions to
                                           elastic energy
                                            0
                                  F (    ( xx 2 +  zz 2   xx 1   zz 1 )d 
                                                 Grain    Grain    Grain    Grain
                                            0




                                                        Grain1




                                                                         Grain2
Expand stress in powers of strain:
   (   A1 + B1 2 + ...
  Non-Linear Driving Force
                             15
                                    (GPa)
          Grain2
                             10
          Grain1

                              5

                                                                                    0.06

                              0
                                                                                                    Tension
-0.03     -0.02      -0.01         0.00      0.01   0.02   0.03 0.05
                                                                                                   Compression
                              -5




                                                               Drving Force (GPa)
                                                                                    0.04


                             -10
                                                                                    0.03


                             -15                                                    0.02


                                                                                    0.01


        Implies driving force of form:                                              0.00
                                                                                       0.0000   0.0001    0.0002           0.0003   0.0004   0.0005


 P( 0            ( A1  A2  0 2 + 1 (B1  B2  03 + ...
                  1                                                                                           Strain
                                                                                                                       2




                  2                    3
Driving Force

                                                                                       • Non-linear
                                  800K T
                      0.09
                                  800K C                                                 dependence of
                                  1000K T                                                driving force on
                      0.08
                                  1000K C                                                strain
                      0.07        1200K T
                                  1200K C                                              • Driving forces are
Driving Force (Gpa)




                      0.06        1400K T                                                larger in tension
                                  14000 C
                      0.05
                                                                                         than compression
                                                                                         for same strain (up
                      0.04                                                               to 17% at 0=0.02)
                      0.03
                                                                                       • Compression and
                      0.02                                                               tension give same
                                                                                         driving force at
                      0.01
                                                                                         small strain
                      0.00                                                               (linearity)
                         0.0000    0.0001   0.0002          0.0003   0.0004   0.0005
                                                        2
                                               Strain
Grain Boundary Motion at Zero Strain

                                          75
     Grain Boundary Position (Angstrom)

                                                            1400K
                                                            1200K
                                          70
                                                            800K

                                          65



                                          60



                                          55



                                          50

                                               0        50000       100000     150000   200000   250000
                                                                                 -14
                                                                    Time Steps (10 s)


                                                   • Fluctuations get larger as T ↑
 Steady State Migration – Low Driving Force
                                              60


         Grain boundary position (Angstrom)

                                              55




                                              50




                                              45




                                              40
                                                   0   20000 40000 60000 80000 100000 120000 140000 160000
                                                                                    -14
                                                                       time steps (10 s)
• At high T, fluctuations can be large
• Determine mobility based upon large boundary displacement
                   Velocity vs. Driving Force
                                         800K                                                                     1000K
                                                                                            6
                 4.0

                 3.5                                                                        5
                               Tensile Strain                                                          Tensile Strain
                 3.0           Compressive Strain                                                      Compressive Strain
                                                                                            4
                 2.5




                                                                           Velocity (m/s)
Velocity (m/s)




                 2.0                                                                        3


                 1.5
                                                                                            2
                 1.0

                                                                                            1
                 0.5

                 0.0
                                                                                            0
                 -0.5
                        0.00    0.01      0.02      0.03     0.04   0.05                        0.00       0.01       0.02       0.03   0.04   0.05
                                       Driving Force (GPa)                                                        Driving Force (GPa)
                     Velocity vs. Driving Force (Continued)
                                         1200K                                                                            1400K
                 6
                                                                                             8

                 5                                                                           7
                             Tensile Strain                                                              Tensile Strain
                             Compressive Strain                                              6           Compressive Strain
                 4
                                                                                             5




                                                                            Velocity (m/s)
Velocity (m/s)




                 3                                                                           4

                                                                                             3
                 2
                                                                                             2

                 1                                                                           1

                                                                                             0
                 0
                                                                                             -1
                      0.00       0.01       0.02       0.03   0.04   0.05                         0.00        0.01           0.02          0.03   0.04
                                        Driving Force (GPa)                                                          Driving Force (GPa)


                     • Velocity under tension is larger than under compression
                       (even after we account for elastic non-linearity)
                     • Difference decreases as T ↑
Determination of Mobility

                 v               120
      M  lim                   110
           p 0  p T             100
                                                Tensile Strain
                                                Compressive Strain
                                    90

                                    80
v/p




                                    70

                                    60




                              v/p
                                    50

                                    40

                                    30

                                    20

                                    10

                                     0

                          p          0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045 0.050
                                                                     p
Activation Energy for GB Migration

                                                                                • Activation energy
       -15.6


       -15.8                                                                      for GB migration is
       -16.0
                                                                                  ~ 0.2 ±0.016eV

       -16.2
ln M




       -16.4


       -16.6


       -16.8


       -17.0
               0.0007   0.0008   0.0009    0.0010    0.0011   0.0012   0.0013
                                                -1
                                          1/T (K )
Conclusion

• Developed new method that allows for the accurate
  determination of grain boundary mobility as a function of
  misorientation, inclination and temperature
• Activation energy for grain boundary migration is finite; grain
  boundary motion is a thermally activated process
• Activation energy is much smaller than found in experiment
  (present results 0.2 eV in Ni, experiment 2-3 eV in Al)
• The relation between driving force and applied strain2 and the
  relation between velocity and driving force are all non-linear
• Why is velocity larger at large strain larger in tension than
  compression?

				
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posted:3/2/2013
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