Appendix L




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Appendix L


Estimation of Boundary Torque and Surface Energy Anisotropy


In normal grain growth, both grain boundary torque and surface energy anisotropy may supply forces for boundary migration in addition to the capillary driving force. The grain boundary torque, , is defined as the derivative of boundary free energy g with respect to its orientation, , i.e.


, (1)



where is the angle of deviation of the boundary from the minimum energy orientation. For convenience, we analyze the normalized boundary torque, i.e. the ratio of boundary torque and the boundary energy here,


. (2)


Similar to the boundary energy anisotropy, surface free energy is also orientation dependent, and may be another source of driving force for boundary migration. For

convenience, we define the normalized surface energy anisotropy as , where S is the surface energy difference.


By utilizing the method proposed by Miller and Williams (1967), the total energy of a system shown in Fig. L-1is minimized when the following results hold


. (3)






L






S


Fig. L-1 Cross section of a grain boundary geometry under the action of boundary torque and surface energy anisotropy in a foil sample.



For each pair of grains, the normalized boundary energy anisotropy, , and the normalized surface energy anisotropy, , can be determined by solving equation (3). Further, the application of these equations to a large number of grain pairs allows the determination of both the frequency of boundary torques and the frequency of surface energy anisotropy in a specific specimen. The results of inclination measurements at 178 pairs of grains in the aluminum foil sample mentioned in section 5.2 are shown in Figs. L-2 and L-3.







Fig L-2. Distribution of normalized boundary torque, /g, of grain boundaries in an Al foil.




Fig L-3. Distribution of normalized surface energy anisotropy, S/g, of grain boundaries in an Al foil.






Fig L-3. Distribution of normalized anisotropy of surface energy in an Al foil.

Appendix M


D
istribution of Dihedral Angles, Curvatures, and CSL Characters of the Grain Boundaries in an Al Foil



Fig. M-1 Distribution of dihedral angles of the grain boundaries in an Al foil. Total number of boundaries is 366.



Fig. M-2 Distribution of curvatures of the grain boundaries in an Al foil. Total number of boundaries is 366.






Fig. M-3 Distribution of CSL grain boundaries in the Al foil, based on Brandon’s criterion, showing large fraction of 1, 3, and 11 boundaries. Total number of boundaries is 366.





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