On grain and subgrain rotations in two dimensions
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I.
INTRODUCTION
G R A I N boundaries and sub-boundaries are interfaces with a higher energy state than a perfect crystallographic lattice. During annealing at elevated temperatures, the total grain boundary and sub-boundary energies can be reduced in several processes. Normal grain growth (cf., for example, References 1 through 10) enables the reduction of the total energy by migration of grain boundaries toward their centers of curvatures. A coarsening of the mean grain size,/9, takes place as a consequence of a reduction of the sizes of the small grains until they are eliminated. It has been suggested that sub-boundaries can migrate toward their centers of curvature in a similar manner as grain boundaries migrate in normal grain growth, m] The energy of the metal can also be reduced by a rotation of the grains or subgrains. The energy of a given boundary will depend on the angles of misorientation. If two neighboring lattices are enabled to rotate in order to eliminate the angles of misorientation, coalescence will take place. The energy of the boundary between the two lattices can thus be released. Hence, rotations of grains and subgrains can yield a reduction in the total energy of a material and will result in a coarsening of the grain or subgrain structure due to coalescence of grains or subgrains. Energy reduction can thus be obtained either by a decrease in the total area of the interfaces, by a decrease in the specific interface energies, or by a combination of both: ( .Yi(iy)Ol-l-i ~, (~y),) ~- Z
6E = Z i
(Ai(if) 6yi(if) )
[1]
i
The first experimental evidence of subgrain coalescence was observed by Fujita t12] and Hu. H334] From a study of the annealing behavior of cold-rolled silicon-iron single crystals, Hu observed that recrystallization nuclei were formed by coalescence of the subgrains in the transition T.O. SAETRE, formerly with the Hydro Aluminum a.s. R&D Center, Karm6y, Norway, is with the Department of Material Science and Engineering, Agder College, N-4890 Grimstad, Norway. N. RYUM is with the Department of Metallurgy, the Norwegian Institute of Technology, 7034 Trondheim, Norway. Manuscript submitted March 14, 1994. METALLURGICAL AND MATERIALS TRANSACTIONS A
band region, where the lattice curvature and dislocation density were high. Hu also proposed a model for nucleation by subgrain coalescence. I~51Based upon Hu's observations, Li t~6] worked out a thermodynamic analysis of subgrain coalescence, which showed that the mechanism of coalescence involves the movement of dislocations from a disappearing low-angle sub-boundary to an adjacent higher angle boundary in which the individual dislocation will have a lower energy. Li also studied the kinetics involved and concluded that either the cooperative movement of dislocations in the boundary or the cooperative diffusion of vacancies in the lattice will be rate controlling. Clear evidence of subgrain coalescence as a mechanism for nucleation of recrystallization in aluminum was found by Faivre and Doherty. t17] They reported that nucleatio
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