Determination of Grain Boundary Stiffness Hao Zhang 1, Mikhail Mendelev 1,2 and David Srolovitz 1 1 PRISM, Princeton University 2 Ames Laboratory

Driving Force X Y Z Grain Boundary Free Surface Grain 2 Grain   5 (001) tilt 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,  xx =  yy =  0 free surface normal to z   iz = 0 Driving Force based on linear Elasticity

Real Driving Force Grain1Grain2 Typical strains 1-2%, out of linear region 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 Implies driving force of form: Zhang H, Mendelev MI, Srolovitz DJ. Acta Mater 52:2569 (2004)

Symmetric boundary  Asymmetric boundary  = 14.04º Asymmetric boundary  = 26.57º  Simulation / Bicrystal Geometry [010]  º

Initial Simulation Cell for Different Inclinations

Mobility vs. Inclinations No mobility data available at  =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

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, ? ? ?

O r n   Determination of Grain Boundary Stiffness Capillarity driven migration Determine reduced mobility from simulation of shrinking, grain Radial velocity for arbitrary curve

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

Circular Shrinkage Geometry

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

 ±0.003 Circular Shape  is 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

Stiffness vs. 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 Using M from the flat boundary simulations and M* from the shrinking grain simulations, we determine stiffness vs. boundary inclination

Conclusion Developed new method (stress driven GB motion) to determine grain boundary mobility as a function of ,  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