Grain boundary mobility during recrystallization of copper
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I.
BACKGROUND
THE migration of grain boundaries is a vital feature of the industrially important annealing processes of recrystallization and grain growth. When considered from a theoretical viewpoint, the velocity of grain boundary migration, v, is generally regarded to be the product of a mobility term, M, and a driving force, Df. Such a linear relationship is obtained whether the theoretical model is based on an absolute reaction rate premise (e.g., Turnbull[1]) or on an irreversible thermodynamics principle (e.g., Machlin[2]). Thus, v 5 M z Df
[1]
has come to be regarded by many as the fundamental relationship for describing mechanistically the grain boundary migration rate during annealing. There has been some experimentally based research, however, taking issue with whether this expression is valid in a general way. Nonlinear velocity/driving force relationships have been proposed in some cases (e.g., Rath and Hu[3]). Solute impurity interactions with moving grain boundaries may lead to deviations from linearity, as has been noted theoretically by Cahn.[4] Recently, in a study of the migration of curvature-driven, single boundaries in high-purity aluminum, where the driving force is relatively small, Gottstein and Shvindlerman[5] demonstrated the validity of the linear expression. They also proved that effects that may appear to be nonlinear can arise due to pinning interactions, e.g., boundary grooving at the free surface, and do not reflect a fundamental change in the mechanism of grain boundary migration or a violation of Eq. [1]. Furthermore, grain growth in pure materials[6] and in materials where pinning forces are accounted
R.A. VANDERMEER, Branch Consultant, is with the Physical Metallurgy Branch, Naval Research Laboratory, Washington, DC 203755343. D. JUUL JENSEN, Senior Scientist, is with the Materials Department, Risø National Laboratory, DK-4000 Roskilde, Denmark. E. WOLDT, Senior Scientist, is with the Institut fu¨r Werkstoffe, Technische Universita¨t Braunschweig, D-38106 Braunschweig, Germany. Manuscript submitted May 16, 1996. METALLURGICAL AND MATERIALS TRANSACTIONS A
for adequately also attests implicitly to the fundamental nature of the linear relationship. For recrystallization, on the other hand, the driving force for grain boundary migration, i.e., the stored energy of cold work, is much larger than the driving force in grain growth. In this case, there is a paucity of work on the velocity vs driving force relationship.[7] Experimentally, it is often observed during recrystallization that the grain boundary migration rate (often called the growth rate) decreases with annealing time. Since English and Backofen,[8] there are many examples where the time dependence of the average grain boundary migration rate, ^v&, e.g., the interface-averaged Cahn and Hagel velocity,[9] can be approximated empirically by ^v& 5 Kch z t 2a
[2]
where Kch and a are constants. Rationalization of a decreasing interface migration rate during recrystallization within the context of the fundamental velocity
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