1. Princeton University Department of Mechanical and Aerospace Engineering The Effect of Inclination on Grain Boundary Mobility Hao Zhang; Mikhail I. Mendelev; David J. Srolovitz

2. 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

3. 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 Grain 2 Grain 1 11 22 33 11 22 33 

4. Steady State Grain Boundary Migration

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

6. Non-Linear Stress-Strain Response ε σ ε*ε* Grain1Grain2 Typical strains as large as 4% (Schönfelder et al.) 1-2% here Measuring Driving force Apply strain ε xx =ε yy =ε 0 and σ zz =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

7. Non-Linear Driving Force Implies driving force of form: Non-linear dependence of driving force on strain 2 Driving forces are larger in tension than compression for same strain Compression and tension give same driving force at small strain (linearity)

8. Velocity vs. Driving Force Velocity under tension is larger than under compression (even after we account for elastic non-linearity) Difference decreases as T ↑ 800K 1200K 1400K 1000K

9. Determination of Mobility p v/p Determine mobility by extrapolation to zero driving force Tension (compression) data approaches from above (below)

10. 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

11. [010]  5 36.87º Symmetric boundary  Asymmetric boundary  = 14.04º Asymmetric boundary  = 26.57º  Simulation / Bicrystal Geometry All simulations performed at fixed misorientation at 1200K

12. 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) (001) (103) (101)

13. GB Diffusivity Dependence on Inclination So far, there is no strong correlation between grain boundary diffusivity and mobility

14. Capillarity Driven Grain Boundary Motion capillarity-driven migration FCC Nickel  5  Tilt Grain Boundary Voter-Chen EAM – Ni w Extract reduced mobility from the rate of change of half-loop volume

15. 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

16. 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 0.75T m Minima in boundary mobility occurs where at least one boundary plane is a low index plane Correlation between grain boundary diffusivity and mobility? Activation energies for grain boundary migration obtained by stress and capillarity driven are similar

18. Outline Motivation Stress Driven Grain Boundary Motion Elastic Driving Force Simulation Method Simulation Results Temperature Effect Inclination Effect Comparison with Capillarity-Driven Motion & Experiment Correlation with Grain Boundary Diffusivity