Static recrystallization kinetics with homogeneous and heterogeneous nucleation using a cellular automata model
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I.
INTRODUCTION
THE kinetics of static recrystallization in cold-worked metals and alloys have traditionally been analyzed using the classical relationships developed by Johnson and Mehl,[1] Avrami,[2] and Kolmogorov[3] (JMAK) and are represented by the Avrami equation[2] F 5 1 2 exp [2B z t k]
[1]
where F is the volume fraction recrystallized, t is the time, and B and k are the model-dependent constants. A plot of log (2ln (1 2 F)) vs log (t) yields a straight line of slope k. The JMAK theory is based on the following assumptions: random and homogeneous nucleation, constant and uniform growth rate, uniform impingement, and spherical grain shapes. The model constants depend on the type of nucleation, i.e., whether it is continuous nucleation, site-saturated nucleation (a fixed number of initial nuclei and a zero nucleation rate at t . 0), or continuously varying nucleation rates. Continuous nucleation becomes site-saturated over time if the growing nuclei consume future nucleation sites. The value of constant k is 3 for site-saturated nucleation with three-dimensional (3-D) growth and 2 for two-dimensional (2-D) growth. For a constant nucleation rate, the value of k is 4 for 3-D growth and 3 for 2-D growth. Cahn[4] extended the JMAK theory to include heterogeneous sitesaturated and constant-rate nucleation at grain boundaries, edges, and corners, resulting in a k value of 1 for boundary nucleation, 2 for edges, and 3 for corners. The work of Vandermeer and Gordon[5] on the recrystallization of zonerefined aluminum correlated with Cahn’s site-saturated edge nucleation, with a k value of 2. Typically, recrystallization data from experimental stud-
R.L. GOETZ, Research Engineer, and V. SEETHARAMAN, Senior Scientist, are with the Materials and Processes Division, UES, Inc., Dayton, OH 45432-1894. Manuscript submitted July 28, 1997. METALLURGICAL AND MATERIALS TRANSACTIONS A
ies are presented in the form of log (2ln (1 2 F)) vs log (t) plots, in expectation of correlating with the JMAK theory. However, experimental results often deviate from the JMAK theory, indicated by nonlinear plots of log (2ln (1 2 F)) vs log (t) or by very low values of k.[6–10] In light of these deviations, several researchers[8–24] have investigated different aspects of static recrystallization from both theoretical and experimental viewpoints. In particular, space and/or time dependencies of the nucleation and growth processes have been examined in detail. Heterogeneous nucleation, nonuniform growth rates, and nonconstant growth rates have been found to cause systematic deviations from the JMAK theory. The role of nonuniform energy storage has also been investigated by Rollet et al.[8] and Hesselbarth et al.[10] Vandermeer and co-workers[13–19] have developed an elegant method known as the microstructural path method (MPM), using Laplace transforms to derive limiting equations for nucleation and growth transformations. The MPM relates the interfacial area per unit volume between the recrystallized and the parent material (SV) to the volume
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