1. Stress Driven Migration of Flat Grain Boundaries Hao Zhang, Mikhail I. Mendelev and David J. Srolovitz Princeton University

2. 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

3. Motivation Want to extract grain boundary mobility from atomistic simulations Half-Loop models are useful, but not sufficient yields the reduced mobility M * =M   ”) not M itself boundary stiffness  ” is difficult to accurately determine from atomistic simulations reduced mobility is a average value over all inclinations Flat boundary geometry can be used to directly determine mobility ( Schönfelder, et al.)

4. 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

5. How Do We Apply a Driving Force? Want constant driving force during simulation avoid NEMD no boundary sliding single 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 Apply strain apply biaxial strain in x and z, free surface normal to y X Z Y Grain Boundary Free Surface Grain 2 Grain 1 11 22 33 11 22 33 

6. Steady State Grain Boundary Migration

7. Driving Force Need accurate determination of driving force Non-symmetric tilt boundary [001] tilt axis boundary plane (lower grain) is (010) Present case:  5 (36.8º) Strain energy density determine using linear elasticity X Z Y Grain Boundary Free Surface Grain 2 Grain 1 11 22 33 11 22 33 

8. Non-Linear Stress-Strain Response ε σ Expand stress in powers of strain: ε*ε* Strain energy density Apply strain ε xx =ε zz =ε 0 and σ yy =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 Grain1Grain2 Typical strains as large as 4% (Schönfelder et al.) 1-2% here

9. Non-Linear Driving Force Implies driving force of form: Drving Force (GPa)

10. Driving Force Non-linear dependence of driving force on strain Driving forces are larger in tension than compression for same strain (up to 17% at  0 =0.02) Compression and tension give same driving force at small strain (linearity)

11. Grain Boundary Motion at Zero Strain Fluctuations get larger as T ↑

12. Steady State Migration – Low Driving Force At high T, fluctuations can be large Determine mobility based upon large boundary displacement

13. Velocity vs. Driving Force 1000K800K Driving Force (GPa)

14. Velocity vs. Driving Force (Continued) Velocity under tension is larger than under compression (even after we account for elastic non-linearity) Difference decreases as T ↑ 1200K Driving Force (GPa) 1400K Driving Force (GPa)

15. Determination of Mobility p v/p

16. Activation Energy for GB Migration Activation energy for GB migration is ~ 0.2 ±0.016eV

17. 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 strain 2 and the relation between velocity and driving force are all non-linear Why is velocity larger at large strain larger in tension than compression?