Dependence of Grain Boundary Mobility on Boundary Plane

1. Dependence of Grain Boundary Mobility on Boundary Plane Hao Zhang1, Mikhail Mendelev1,2 and David Srolovitz1 1PRISM, Princeton University 2Ames Laboratory

2. Challenges • Neither curvature driven boundary migration experiments nor simulations yield the fundamental kinetic properties for grain boundary migration • ,M* is the product of the mobility and grain boundary stiffness • Reduced mobility is averaged over all possible inclinations • The migration of a flat boundary is easier to analyze, but has several limitations • Can yield grain boundary mobility dependence on inclination • Is the variation of grain boundary mobility correlated with other boundary properties, such as grain boundary energy and self-diffusivity?

3. Free Surface 11 22 33 q Grain 2 22 11 33 Grain Boundary Z Grain 1 X Y Free Surface Elastically-Driven Migration of a Flat 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 • Applied strain • constant biaxial strain in x and y • free surface normal to z  iz = 0 • Driving Force based on linear Elasticity S5 (001) tilt boundary

4. Grain1 Grain2 Measured Driving Force • Typical strains • 1-2%, out of linear region • Implies driving force of form: • Measuring driving force • Apply strain εxx=εyy=ε0and σiz= 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

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

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

7. Initial Simulation Cell for Different Inclinations

8. Mobility vs. Inclination • 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

9. Activation Energy vs. Inclination • The variation of activation energy for grain boundary migration over the inclination region we studied is significant • The variation of mobility becomes weaker than expected on the basis of activation energy because of the compensation effect • Activation energy for the symmetric boundary is unknown

10. Diffusivity vs. Inclination • Diffusivity shows more anisotropic at low temperature than at high temperature • Most of local minimum corresponds to one of the grains normal with low Miller indices • The a=0º has a change from minimum to maximum

11. Activation Energy and Compensation Effect • The activation energy all lie between 0.5 to 0.6 eV, except for the a=0º symmetric boundary(1.1 eV) • Compensation effect weaken the diffusivity variation based upon the activation energy for self-diffusion

12. Mobility, Self-diffusion and Energy • At low temperature, self-diffusion and grain boundary energy have similar trend, i.e. change from minimum to maximum, but mobility has opposite trend. • Mobility, self-diffusion coefficient and grain boundary energy shows local minimum at special inclination (one of the plane normal is low Miller indices) • There exists correlation between those three quantities in the inclination range of 18º to 45º.

13. Conclusion • Used stress driven GB motion to determine grain boundary mobility as a function of q, a and T • Mobility is a strong function of inclination and temperature • Grain boundary self-diffusion is sensitive to inclinations, i.e. grain boundary structure • Minima in boundary mobility, self-diffusion coefficient and grain boundary energy occurs where at least one boundary plane is a low index plane • In the inclination range from 18º to 45º, there is a strong correlation between grain boundary diffusivity, energy and mobility