Determination of Grain Boundary Stiffness

1. Determination of Grain Boundary Stiffness Hao Zhang1, Mikhail Mendelev1,2 and David Srolovitz1 1PRISM, Princeton University 2Ames Laboratory

2. Free Surface 11 22 33 q Grain 2 22 11 33 Grain Boundary Z Grain 1 X Y Free Surface Driving Force • 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, exx = eyy= e0 • free surface normal to z  iz = 0 • Driving Force based on linear Elasticity S5 (001) tilt boundary

3. Grain1 Grain2 Real Driving Force • Typical strains • 1-2%, out of linear region • Implies driving force of form: • Measuring driving force • Apply strain εxx=εyy=ε0and 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: • Driving force Zhang H, Mendelev MI, Srolovitz DJ. Acta Mater 52:2569 (2004)

4. a a Symmetric boundary Asymmetric boundary a = 14.04º Asymmetric boundary a = 26.57º Simulation / Bicrystal Geometry [010] S5 36.87º

5. Initial Simulation Cell for Different Inclinations

6. Mobility vs. Inclinations • No mobility data available at a=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

7. Activation Energy vs. Inclination • The variation of activation energy for migration with inclination is significant • The variation of mobility is weaker than expected on the basis of activation energy because of the compensation effect • Activation energy for symmetric boundaries, ? ? ?

8. <010> <010> <100> <100> r n a q O Determination of Grain Boundary Stiffness • Determine reduced mobility from simulation of shrinking, grain • Capillarity driven migration • Radial velocity for arbitrary curve

9. r n a q O Determination of Grain Boundary Stiffness (Contd) • If grain shape is only slightly different from a circle, we can assume 4-fold symmetry - [010] tilt • Substituting into expression for the velocity and rearranging terms • Keep the first order terms • To find how the reduced mobility varies with inclination, a, we must relate atoq

10. Circular Shrinkage Geometry

11. Simulation Result • Steady-state migration during circular shrinkage • Migration velocity strongly depends on temperature • Activation energy for migration is 0.2eV

12. Circular Shape d=-0.013±0.003 dis temperature independent between 1000 and 1400 K to within the accuracy of these simulations assumed functional form of grain shape is in agreement with simulation results

13. Stiffness vs. Inclination Using M from the flat boundary simulations and M* from the shrinking grain simulations, we determine stiffness vs. boundary inclination • At high temperature, Stiffness is not significantly changed with inclinations • General speaking, stiffness is larger at low T than at high T • The ratio of maximum to minimum at 1000K is ~3 • Can not determine the existence of cups around the two symmetric grain boundaries

14. Conclusion • Developed new method (stress driven GB motion) to determine grain boundary mobility as a function of q, a and T • Extracted grain boundary stiffness from atomistic simulations • Mobility is a strong function of inclination and temperature; mobility exhibits minima where at least one of the boundary planes has low Miller indices • Grain boundary stiffness varies with inclination and is only weakly temperature-dependent